1. Computational and Systems Biology
  2. Neuroscience
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A unified computational model for cortical post-synaptic plasticity

  1. Tuomo Mäki-Marttunen  Is a corresponding author
  2. Nicolangelo Iannella
  3. Andrew G Edwards
  4. Gaute T Einevoll
  5. Kim T Blackwell
  1. Simula Research Laboratory, Norway
  2. Department of Biosciences, University of Oslo, Norway
  3. Faculty of Science and Technology, Norwegian University of Life Sciences, Norway
  4. Department of Physics, University of Oslo, Norway
  5. The Krasnow Institute for Advanced Study, George Mason University, United States
Research Article
Cite this article as: eLife 2020;9:e55714 doi: 10.7554/eLife.55714
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11 figures and 4 tables

Figures

Signalling pathways included in the model.

The PKA-pathway-related proteins and signalling molecules are highlighted by blue, PKC-pathway molecules by yellow, and CaMKII-pathway molecules by green colours. Reactions associated with a molecular species in parenthesis indicate a dependency on the denoted species — for details, see Table 3. Acronyms: β-AR – β-adrenergic receptor; AC1 and AC8 – adenylyl cyclase type 1 or 8; CaM – calmodulin; CaMKII – calmodulin-dependent protein kinase II; cAMP – cyclic adenosine monophosphate; DAG – diacylglycerol; Epac1 – exchange factor directly activated by cAMP 1; Gi, Gq and Gs – G-protein type I, Q, or S; GluR1 and GluR2 – AMPAR subunit 1 or 2; mGluR – metabotropic glutamate receptor; M1R – cholinergic receptor M1; NCX – Na+-Ca2+ exchanger; Ng – neurogranin; NMDAR – NMDA receptor; PDE1 and PDE4 – phosphodiesterase type 1 or 4; PIP2 – phosphatidylinositol 4;5-bisphosphate; PKA – protein kinase A; PKCt and PKCp – transiently or persistently active protein kinase C; PLC – phospholipase C; PMCA – plasma membrane Ca2+ ATPase; PP1 – protein phosphatase 1; PP2A – protein phosphatase 2A; PP2B – protein phosphatase 2B (calcineurin). In this work, the NMDARs are considered only in section 'Paired pre- and post-synaptic stimulation induces PKA- and PKC-dependent spike-timing-dependent plasticity (STDP) in GluR1-GluR2-balanced synapses': in the rest of the work, Ca2+ is directly injected as a square-pulse current into the spine.

Ca2+ activates CaMKII, PKA, and PKC pathways.

(A) Illustration of the stimulus protocols with Ca2+ flux amplitudes 150 (green), 200 (cyan), and 250 (purple) particles/ms. (B–F) Time courses of Ca2+ (in nM) bound to buffers (B), pumps (C), PKC-pathway proteins (D), or CaM (E), and the concentration of free Ca2+ ions (F), according to NeuroRD (solid; averaged across eight samples) or NEURON (dashed) simulations. Colours indicate the Ca2+ flux used (see A). (B) Number of Ca2+ ions bound to Ca2+ buffers, that is immobile buffer and calbindin. (C) Number of Ca2+ ions bound to Ca2+ pumps and exchangers, that is PMCA and NCX. (D) Number of Ca2+ ions bound to PKC-pathway proteins PLC and PLA2. (E) Number of Ca2+ ions bound to CaM, in all its forms. (F) Cytosolic Ca2+ concentration (mM) (G) Degrees of activation of different Ca2+-binding proteins in a steady state (5 min after onset of Ca2+ input) as a function of the magnitude of Ca2+ flux. The x-axis shows the amplitude of the Ca2+ input (see panel A), and the y-axis shows the ratio of the underlying species in a Ca2+-bound form over the total number of the proteins. For CaM, only the CaM molecules bound by four Ca2+ ions are considered activated — in PLC, PLA2, and DGL, binding of only one Ca2+ ion is needed for activation. Here, the measured quantity of active PLC includes both Gq-bound and non-Gq-bound CaPLC. Inset: zoomed-in view on the red area. (H) Ratio of the steady-state concentration of PKA catalytic subunit over the theoretical maximum where all PKA molecules were dissociated into residuals and catalytic subunits. Colour of the curve indicates the amplitude of the β-adrenergic ligand flux (particles/ms). (I) Fraction of phosphorylated CaMKII subunits. (J) Fraction of (transiently or persistently) activated PKC. Colour of the curve indicates the amplitude of the cholinergic and glutamatergic ligand flux (particles/ms). The grey area in panels (GJ) represents Ca2+ inputs that cause cytosolic Ca2+ concentration to reach extremely high levels (>1 mM) that are likely to lead to apoptosis.

Figure 3 with 4 supplements
4xHFS activates CaMKII, PKA, and PKC pathways and leads to LTP (A–R), while LFS activates the PKC pathway and leads to LTD (S–U).

(A) Total synaptic conductance in response to 4xHFS, determined by the numbers of membrane-inserted GluR1s and GluR2s — see Equation 5. The stimulation starts at 40 s and lasts until 53 s. (B–C) Concentration of membrane-inserted GluR1s (B) and GluR2s (C) in response to 4xHFS. (D–H) Concentration of different species in the CaMKII pathway, namely, intracellular unbound Ca2+ (D), CaM bound with two Ca2+ ions (E), CaM bound with four Ca2+ ions (active CaM; F), phosphorylated CaMKII, bound or unbound by CaMCa4 (G), and S831-phosphorylated and double-phosphorylated GluR1 subunits (H) in response to 4xHFS. (I–M) Concentration of different species in the cAMP-PKA pathway, namely, β-adrenergic ligand in all its forms (I), activated (GTP-bound but not bound to ATP) Gs and Gi proteins (J), intracellular cAMP (K), catalytic subunit of PKA (L), and S845-phosphorylated and double-phosphorylated GluR1 subunits (M) in response to 4xHFS. (N–R) Concentration of different species in the PLC-PKC pathway, namely, glutamate and acetylcholine in all their forms (N), activated (GTP-bound but not bound to DAG) Gq proteins (O), intracellular DAG (P), activated PKC (Q), and S880-phosphorylated GluR2 subunits (R) in response to 4xHFS. S: Total synaptic conductance in response to LFS. (T–U) Concentration of membrane-inserted GluR1s (T) and GluR2s (U) in response to LFS, which starts at 40 s and lasts until 220 s. The solid lines represent stochastic (NeuroRD) simulation results, while the dashed lines represent data from deterministic (NEURON RxD) simulations. β-adrenergic ligands, glutamate, and acetylcholine are measured in numbers of particles as they reside both at the membrane (when bound to receptors) and at the extracellular subspace near the spine membrane (when unbound); other species measured in concentration.

Figure 3—figure supplement 1
Both GluR1 and GluR2 are needed for bidirectional plasticity.

(A) When GluR1 subunits are absent, 4xHFS induces LTD instead of LTP. (B) When GluR2 subunits are absent, LFS does not lead to changes in synaptic conductance. (C) When GluR1 subunits are absent, the paired-stimulus protocol induces acetylcholine-mediated LTD but not β-adrenergic ligand-mediated pairing interval-dependent LTP. (D) When GluR2 subunits are absent, the paired-stimulus protocol induces β-adrenergic ligand-mediated pairing interval-dependent LTP, but not acetylcholine-dependent LTD. (A–B) The y-axis shows the total synaptic conductance, and the x-axis shows the time. The 4xHFS stimulation lasts from 40 to 53 s, and the LFS stimulation from 40 to 220 s. See Figure 3 for details. (C–D) The y-axis shows the relative synaptic conductance, and the x-axis shows the inter-stimulus interval (ISI). See Figure 5 for details.

Figure 3—figure supplement 2
An alternative dimers-of-like-dimers rule of tetramer formation reproduces the HFS-induced LTP, LFS-induced LTD, and STDP predictions obtained with the default tetramer formation rule.

(A) Plasticity induced by HFS and LFS. The y-axis shows the total synaptic conductance, and the x-axis shows the time. The 4xHFS stimulation lasts from 40 to 53 s, and the LFS stimulation from 40 to 220 s. See Figure 3 for details. (B) Plasticity induced by pairing protocol in the presence of various neuromodulators. The y-axis shows the relative synaptic conductance, and the x-axis shows the inter-stimulus interval (ISI). See Figure 5 for details. In all these experiments, a dimers-of-like-dimers rule of tetramer formation was used. 35% of the GluR subunits were GluR1 type and 65% GluR2.

Figure 3—figure supplement 3
The biochemical signalling network model, given the NMDAR-conducted Ca2+ inputs from the multicompartmental neuron model of layer 2/3 pyramidal cell under 1.3 mM extracellular [Mg2+], predicts LTP for 6xHFSt and LTD for LFS-1Hz.

The 6xHFSt protocol lasts from 0 to 60 s, while the LFS-1Hz protocol lasts from 0 to 1800s (data shown until 500 s). See Materials and methods, section 'Modelling the Ca2+ inputs and neuromodulatory inputs' for details on the stimulation protocols and the multicompartmental neuron model. All initial concentrations of the biochemical model were the same as in Figure 3.

Figure 3—figure supplement 4
The biochemical signalling network model robustly predicts LTP for HFS and LTD for LTP with altered durations of neuromodulatory inputs.

In the control case, the pre-synaptic input-associated fluxes of β-adrenergic and cholinergic ligands were 10 and 20 particles/ms, respectively, and the duration of each pulse was 3 ms. Here, the ligand inputs were made four times steeper by reducing the duration to 0.75 ms and the increasing the amplitudes to 40 (β-adrenergic) and 80 (cholinergic) particles/ms (blue), or 3.2 times less steep by increasing the duration to 9.6 ms and reducing the amplitudes to 3.125 (β-adrenergic) and 6.25 (cholinergic) particles/ms (green). Both alterations gave LTP and LTD responses that were indistinguishable from those of the control case. The red dashed line shows the LTP and LTD responses where the pre-synaptic input-associated fluxes of β-adrenergic and cholinergic ligands were replaced by a 10 min pulses of amplitudes 0.4 (β-adrenergic) and 0.8 (cholinergic) particles/ms, respectively, initiated 6 min prior to the pre-synaptic inputs to mimic the bath application of corresponding agents.

4xHFS-induced LTP is dependent on β-adrenergic ligands and LFS-induced LTD is dependent on activation of mGluRs or cholinergic receptors.

(A–D) 4xHFS-induced LTP in the control case (dark purple), without Ca2+ inputs (blue), without β-adrenergic ligands (green), and under blockade of PKC pathway-activation (mGluRs or cholinergic receptors; yellow). (E–H) LFS-induced LTD in the control case (dark purple), under the blockade of mGluR activation (blue), and under blockade of both mGluRs or cholinergic receptors (yellow). (A, E) Total synaptic conductance. (B, F) Membrane expression of GluR1. (C, G) Membrane expression of GluR2. (D, H) The fraction of membrane-inserted GluR1 over all membrane-inserted GluR subunits (left), the probability of an AMPAR tetramer being homomeric GluR1 (middle), and the relative contribution of homomeric GluR1 subunits to the total conductance (i.e., summed conductance of homomeric GluR1 tetramers divided by the summed conductance of all tetramers; right). The bars represent the values at the end of the 4xHFS (D) or LFS (H) simulation with (dark purple) and without (yellow) PLC-activating ligands.

Figure 5 with 1 supplement
Layer 2/3 pyramidal cell plasticity in response to STDP protocol depends on neuromodulatory state and pairing interval.

(A) Layer 2/3 pyramidal cell morphology (grey, thin), locations of synaptic input highlighted (black, thick). Inset: Illustration of the inputs (black) and the recorded synaptic intracellular Ca2+ (grey). Scale bar 200 μm. (B–D) Membrane potential at the dendritic spine when the pre-synaptic stimulation onset is 50 ms after (B), at the same time as (C), or 50 ms prior to (D) the onset of the last somatic stimulus. Inset (red): Mg2+-gate variable as a function of time, ranging from −80 ms to 140 ms in a similar manner as the data in the main panel. (E–G) Concentration of free Ca2+ in the dendritic spine according to the biochemical spine model when the pre-synaptic stimulation onset is 50 ms after (B), at the same time as (C), or 50 ms prior to (D) the onset of the last somatic stimulus. (H–J) No LTD was induced by the stimulation protocol (1 Hz paired with post-synaptic stimulation for 2 min) in the absence of M1-receptor activation, but pairing-interval-dependent LTP was induced in presence of β-adrenergic inputs. (K–M) Pairing-interval-dependent LTD was induced when the synaptic input was coupled with cholinergic inputs, and STDP was induced when both cholinergic and β-adrenergic inputs were present. (H, K) Relative concentration of GluR1 at the membrane 16 min after the stimulation onset (normalised by concentration of membrane-inserted GluR1 at rest). (I, L) Relative concentration of GluR2 at the membrane 16 min after the stimulation onset (normalised by concentration of membrane-inserted GluR2 at rest). (J, M) Relative synaptic conductance (Equation 5) 16 min after the stimulation onset (normalised by synaptic conductance at rest).

Figure 5—figure supplement 1
Ca2+ fluxes predicted by the multicompartmental layer 2/3 pyramidal cell model depend on the inter-stimulus interval (ISI).

(A–D) The Ca2+ flux time course during the pairing protocol (from 100 ms before to 900 ms after the pre-synaptic stimulus) entering the post-synaptic spine through NMDA receptors when the onset of the last of the four post-synaptic stimuli is 50 ms before (A), coincident with (B), 50 ms after (C), or 150 ms after (D) the onset of the pre-synaptic stimulus. (E) Peak Ca2+ flux (from the data of panels A–D) for different ISIs. (F) Average Ca2+ flux (averaged across the time points of panels A–D, i.e., −100 to 900 ms) for different ISIs.

Figure 6 with 1 supplement
The STDP curve of layer 2/3 pyramidal cells is affected by the number of post-synaptic stimulus pulses associated with the pre-synaptic input.

(A) The STDP curves of Figure 5M when the number of spikes per post-synaptic burst was 1 (yellow), 2 (green), 3 (blue), or 4 (as in Figure 5; dark purple). Inset: relative concentrations of membrane-inserted GluR1 (top) or GluR2 (bottom) subunits — see Figure 5K–L for reference. (B) Top: somatic membrane potential time course (aligned according to the onset of the first stimulus) for different numbers of post-synaptic stimulus pulses. Bottom: somatic SK current-density time course in the four conditions. Inset: the SK current densities 200 ms after the onset of the first post-synaptic stimulus. (C–D) Ca2+ flux to the dendritic spine when the pre-synaptic stimulation onset is 200 ms after (C) or 30 ms before (D) the onset of the last post-synaptic stimulus. (E) The STDP curves of Figure 5M when the number of spikes per post-synaptic burst was four but the somatic SK conductance parameter was either normal (dark purple), 50% smaller (magenta), or 80% smaller (pink).

Figure 6—figure supplement 1
The post-STDP synaptic conductance is weakly correlated with the peak of the Ca2+ input but strongly correlated with the mean Ca2+ input during the inter-stimulus interval.

The y-axis shows the post-STDP synaptic conductance (in presence of β-adrenergic and cholinergic neuromodulation) relative to baseline as in Figure 5M. (A) The x-axis shows the peak Ca2+ input (in particles/ms) for different ISIs (see Figure 5D–F and Figure 5—figure supplement 1E). (B) The x-axis shows the time integral of Ca2+ input of Figure 5D–F from −500 to 500 ms, normalized by 1000 ms (see Figure 5—figure supplement 1F). Different colours represent different ISIs between −200 and 200 ms (intervals 0,±10,±20, and ±50 labeled for reference). The blue dashed line shows the linear regression, and the blue text shows the corresponding Pearson correlation coefficient.

Figure 7 with 1 supplement
The fraction of GluR1s, number of Ca2+ extrusion proteins, and the concentrations of PKA and PKC-pathway proteins in the post-synaptic spine determine the type of LTP/LTD in the post-synaptic spine.

(A) The LTP/LTD curves for all 16 classes. Four values of Ca2+ input amplitude were considered: 0, 50, 150, and 250 particles/ms (x-axis; repeated and overlaid for space). The y-axis shows the relative synaptic conductance, that is, total synaptic conductance 15 min after the onset of the Ca2+ input divided by the total synaptic conductance before the Ca2+ input. 20 representative parameter sets are displayed from each class, coloured from purple (lowest relative synaptic conductance response for medium Ca2+ input) to green (highest conductance). The black, dashed trace in class six represents the model with the default concentration parameters. (B–E) Distribution of model parameters, that is, GluR1 ratio (B), NCX-concentration coefficient (C), PKA pathway-concentration coefficient fPKA (D), and PKC pathway-concentration coefficient fPKC in the 16 classes. Class 6 (purple) highlighted for further analysis. F–H: GluR1 ratio plotted against NCX-concentration coefficient (F), fPKA (G), and fPKC (H) in class 6. The contours represent the distribution of parameters (N = 5837) that produced class-6 plasticity. No parameters yielding class-6 plasticity were found beyond the purple contour, and the inner contours cover the parameter space where the distribution is higher than 0%, 20%, 40%, 60% or 80% of the maximal density value. The black and red markers represent parameter sets that produced two plasticity subclasses, namely, one where the total deviance (summed absolute difference) from the BCM-type plasticity produced by the default parameter set (black, N = 145) or from a linearly increasing LTP (red, N = 183) was less than 0.2 (a.u.). Inset: The LTP/LTD plasticity curves of the two subclasses. The thick lines represent the centre of the subclasses (black: relative conductances in response to 50, 150, and 250 Ca2+ ions/ms: 0.76, 0.96, 2.24; red: relative conductances in response to 50, 150, and 250 Ca2+ ions/ms: 1.41, 1.83, 2.24).

Figure 7—figure supplement 1
The PKC-pathway parameter distributions differ between clusters separated by their response to low (50 particles/ms) Ca2+ input.

Here, the experiment of Figure 7 is repeated by clustering the parameters sets based on the relative synaptic conductance after a steady-state Ca2+ flux of 50 particles/ms (250 particles/ms in Figure 7) to highlight the contribution of the PKC-pathway parameter. (A) The LTP/LTD curves for all 16 classes (classes 1’—16’). Four values of Ca2+ input amplitude were considered: 0, 50, 150, and 250 particles/ms (x-axis; repeated and overlaid for space). The y-axis shows the relative synaptic conductance, that is, total synaptic conductance 15 min after the onset of the Ca2+ input divided by the total synaptic conductance before the Ca2+ input. 20 representative parameter sets are displayed from each class, coloured from purple (lowest relative synaptic conductance response for medium Ca2+ input) to green (highest conductance). The black, dashed trace in class 6’ (note that this is a different group than class six in Figure 7) represents the model with the default concentration parameters. (B–E) Distribution of model parameters, that is, GluR1 ratio (B), NCX-concentration coefficient (C), PKA pathway-concentration coefficient fPKA (D), and PKC pathway-concentration coefficient fPKC in the 16 classes. Class 6’ (purple) highlighted for further analysis. (F–H) GluR1 ratio plotted against NCX-concentration coefficient (F), fPKA (G), and fPKC (H) in class 6’. The contours represent the distribution of parameters (N = 7275) that produced class-6’ plasticity. No parameters yielding class-6’ plasticity were found beyond the purple contour, and the inner contours cover the parameter space where the distribution is higher than 0%, 20%, 40%, 60% or 80% of the maximal density value. The black and red markers represent parameter sets that produced two plasticity subclasses, namely, one where the total deviance (summed absolute difference) from the BCM-type plasticity produced by the default parameter set (black, N = 98) or from a linearly decreasing LTP (red, N = 44) was less than 0.2 (a.u.). Inset: The LTP/LTD plasticity curves of the two subclasses. The thick lines represent the centre of the subclasses (black: relative conductances in response to 50, 150, and 250 Ca2+ ions/ms: 0.76, 0.96, 2.24; red: relative conductances in response to 50, 150, and 250 Ca2+ ions/ms: 0.76, 0.52, 0.27).

The model predictions of LTP and LTD are robust to small changes in model parameters.

Values of initial concentrations (47 parameters) or reaction rates (223 parameters) were changed one at the time by −10% or +10%, and the resulting synaptic conductance 16 min after LFS (A) or 4xHFS (B) protocol was measured (NEURON RxD simulations). The initial synaptic conductance is 33.4 pS (see Figure 3A,S), although some parameter changes mildly affected this value (data not shown). The x-axis shows the post-LFS (A) or post-HFS (B) synaptic conductance, and the y-axis shows the number of parameter alterations. Majority of the parameter changes had small effect on plasticity (grey bars), but changes in initial concentrations of 10 species and 12 reaction rates caused >15% change in the amplitude of LTP or LTD — these changes are represented by black (multi-pathway parameters), blue (PKA-pathway-related parameters), and green (PKC-pathway-related parameters) bars. The underlying parameter changes are printed above the corresponding bar.

Figure 9 with 11 supplements
The model can be fit to LTP/LTD data from different cortical areas.

(A) The model could be fit to LTP/LTD data from data sets EC-1 (top), EC-2, PFC-1, PFC-2, BC, ACC, PFC-3, AuC-1, and AuC-2 (bottom). The curves represent the model predictions of the best-fit parameter sets, and the dots represent the experimental data from Table 2. For data sets other than AuC-1 and AuC-2, several experiments with various chemical agents or genetic mutations were performed for each neuron population: these are ordered as in Table 2 (e.g., in data set EC-1, purple (1 st experiment) corresponds to control, blue (2nd experiment) to CaMKII-blocked experiment, and green (3rd experiment) to the experiment where post-synaptic Ca2+ was blocked). (B) The model could not be fit to the complete LTP/LTD data from data sets VC-1 (top) and VC-2 (bottom). The best parameter sets correctly predicted the LTP/LTD in up to two experiments (e.g., the selected parameter sets reproduce the HFS data with and without CaMKII inhibitor, but failed to reproduce the LFS data). (C) The model could be fit to the LTP/LTD data from data sets VC-1 (top) and VC-2 (bottom) when CaMKII-blocked experiments were ignored. The vertical bars in (B) and (C) represent the SD from the experimental data. (D) Proportion of accepted parameter sets across the 20 generations of multi-objective optimisation (20’000 parameter sets in total) in each data set. (E) Box plots of selected parameters in the acceptable parameter sets of data sets EC-1 and EC-2 (three left-most pairs) and AuC-1 and AuC-2 (right-most pair). Values of fCaMKII and fPKA are linearly scaled such that the values 0 and 1 correspond to 0 and double the original value of the underlying parameters, respectively (CaM and CaMKII for fCaMKII, and R, Gs, AC1, and AC8 for fPKA, see Materials and methods, section 'Parameter alterations and model fitting'). The medians were significantly different in the compared data sets (U-test, p-value<0.001).

Figure 9—figure supplement 1
Parameters for data set EC-1.

Ranges of each parameter among the accepted parameter sets from the fitting of Figure 9. All parameters are linearly scaled between 0 and 1, which represent the minimal and maximal value of the corresponding model parameter. The values of the parameters reproducing the data of Figure 9A and C are presented by red dots, and their exact (non-scaled) values are listed below the figure. The values of the error functions corresponding to the listed parameter set are displayed in the right-most column for each experiment.

Figure 9—figure supplement 2
Parameters for data set EC-2.
Figure 9—figure supplement 3
Parameters for data set PFC-1.
Figure 9—figure supplement 4
Parameters for data set PFC-2.
Figure 9—figure supplement 5
Parameters for data set BC.
Figure 9—figure supplement 6
Parameters for data set ACC.
Figure 9—figure supplement 7
Parameters for data set PFC-3.
Figure 9—figure supplement 8
Parameters for data set VC-1.
Figure 9—figure supplement 9
Parameters for data set VC-2.
Figure 9—figure supplement 10
Parameters for data set AC-1.
Figure 9—figure supplement 11
Parameters for data set AC-2.
The models describing plasticity in different cortical areas predict diverse responses to modified stimulation protocol and stimulation under chemical blockers.

(A) The predicted responses of the 20 best models in each data set to HFS (100 pulses at 100 Hz) stimulation. (B–D) The predicted responses of the 20 best models in each data set to the applied stimulation protocol (see Table 2) when CaMKII (B), PKA (C), or PKC (D) activity was blocked (red) or under control condition (black).

Figure 11 with 2 supplements
Calibration of the model.

Black curves represent the final model, while grey lines represent predictions of models where previous model components or tentative parameter values were used. (A) Concentration of membrane-inserted GluR2 in 4xHFS when the forward rate of the membrane insertion of non-phosphorylated GluR2 was 0.0055 1/ms Gallimore et al., 2018 (grey) or 0.00025 1/ms (black). The rate 0.00025 1/ms caused a resting-state concentration of 121 nM for the membrane-bound GluR2 subunits, which is 45% of the total GluR2 concentration (270 nM). (B) Steady-state concentration of activated (bound by four Ca2+ ions) CaM in response to a prolonged Ca2+ input amplitude when the two-step (grey) or three-step (black) activation of CaM by Ca2+ was used. The x-axis shows the corresponding steady-state concentration of free Ca2+. Here, the initial concentrations of molecular species were as in Li et al., 2020, namely, 50 μM for CaM, Ng, PP2B, and CaMKII and 0 μM for all other species. Red dots show experimental data from Hoffman et al., 2014. (C) Concentration time course of non-protein-bound activated CaM (inset) or total activated CaM (main figure) in response to 4xHFS when the two-step (grey) or three-step (black) activation of CaM by Ca2+ was used. (D) Percentage of S880-phosphorylated GluR2 15 min after LFS when different forward rates of the activation of persistent PKC (kf between 0.00005 and 0.005 1/(nM ms)) were used. The value kf = 0.0005 1/(nM ms) gave a percentage of 47%, in close agreement with Ashby et al., 2004. (E–G) The dynamics of transiently active PKC (E) were not strongly influenced by the forward rate of the activation of persistent PKC (reaction 140), but those of persistently active PKC (F) and S880-phosphorylated GluR2 (G) were significantly affected. Black curves show the data corresponding to kf = 0.0005 1/(nM ms), while the grey lines show the data corresponding to kf = 0.00015 1/(nM ms) (dashed) and kf = 0.0015 1/(nM ms) (dotted). (H) Predicted responses of an isolated PKA activation model (reactions 59 and 93) to a 16 s cAMP input (dim grey background) when different values of the forward rate of PKA binding with four cAMP molecules were used. The curves show the concentration of the catalytic PKA subunit when different forward rates of PKA–cAMP binding (from bottom to top: 0.4 ×109, 1.0×10 9, 1.6 ×109, 2.2×10 9, and 2.8 ×109 1/(nM4ms)) were used. The markers show the corresponding data when the two-step PKA–cAMP binding model of Jȩdrzejewska-Szmek et al., 2017 was used. Inset: summed absolute differences between the tentative data (curves) and simulated data from the previous model (markers). The model with the forward rate of kf = 1.6 × 109 1/(nM4ms) gave the closest correspondence to the model of Jȩdrzejewska-Szmek et al., 2017. (I) Concentration of S845-phosphorylated GluR1 in response to 4xHFS when the single-step (reaction 59, black) or two-step (from Jȩdrzejewska-Szmek et al., 2017, grey) PKA–cAMP binding was used. (J) Concentration of S831-phosphorylated GluR1 in response to 4xHFS when PKC did (black) or did not (grey, overlaid) phosphorylated S831 in GluR1s.

Figure 11—figure supplement 1
1 hr of simulation without inputs is sufficient to obtain a steady state.

The model was run for 4040 s without inputs. The absolute values of the time derivatives for each molecular species were determined, and the black curve shows the sum of these derivatives (nM/sec) across the species. The dashed blue line shows the concentration derivative value that corresponds to a change of one molecule/sec in a spine of volume 0.5 μm3.

Figure 11—figure supplement 2
The model STDP model is robust to changes in AMPA conductance but sensitive to changes in NMDA condutance in the multicompartmental layer 2/3 pyramidal cell model.

The model of Figure 5 was simulated with altered AMPAR (A) or NMDAR (B) conductances, and the effects on STDP curves were measured. See Figure 5 for details. (A) Half (blue) or double (red) the original value (black) of the AMPAR conductances were used. (B) Half (blue) or double (red) the original value (black) of the NMDAR conductances were used.

Tables

Table 1
Pathways contributing to cortical synaptic plasticity.

(A) Experimental evidence on the requirement of various molecular species for specific types of synaptic regulation in different cortical areas. (B) Model components needed for describing the modes of plasticity listed in (A). References are made to previous computational models describing these pathways. The types of phosphorylation of AMPAR subunit that mediate the plasticity are printed in bold.

(A)
Pathway componentsType of neuronsType of regulationPre-/post-synapticReferences
CaMKIICingulate cortexEsophageal acid-induced sensitisationpost-syn.Banerjee et al., 2013
CaMKIIPrefrontal cortex, pyramidal neurons5-HT1-induced modulation of AMPA currentspost-syn.Cai et al., 2002
β-adr. receptors, PKAVisual cortex, layer 4 pyramidal cellsPotentiation of AMPA currentspost-syn.Seol et al., 2007
M1 receptors, PKCVisual cortex, layer 4 pyramidal cellsDepression of AMPA currentspost-syn.Seol et al., 2007
D1–PKAPrefrontal cortex, pyramidal neuronsPotentiation of AMPA currentspost-syn.Sun et al., 2005
β-adr. receptorsFrontal cortexPotentiation of field EPSPsn/aSáez-Briones et al., 2015
PKCCultured cortical neuronsInternalisation of AMPARspost-syn.Chung et al., 2000
ERKVisual cortexPotentiation of field EPSPsn/aDi Cristo et al., 2001
(B)
Molecular pathwayCell type and references
Ca2+ → CaM → CaMKIIHippocampal CA1 neuron Bhalla and Iyengar, 1999; Jȩdrzejewska-Szmek et al., 2017, generic Hayer and Bhalla, 2005,
cerebellar Purkinje cells Gallimore et al., 2018, striatal spiny projection neuron Blackwell et al., 2019
CaMKII → GluR1 S831pHippocampal CA1 neuron Jȩdrzejewska-Szmek et al., 2017
β-adrenergic receptors → cAMPHippocampal CA1 neuron Jȩdrzejewska-Szmek et al., 2017
cAMP → PKAHippocampal CA1 neuron Bhalla and Iyengar, 1999; Jȩdrzejewska-Szmek et al., 2017, cerebellar Purkinje
cells Gallimore et al., 2018
PKA → GluR1 S845pHippocampal CA1 neuron Jȩdrzejewska-Szmek et al., 2017
M1 receptors → PLCCerebellar Purkinje cells Gallimore et al., 2018
PLC → PKCHippocampal CA1 neuron Bhalla and Iyengar, 1999, striatal spiny projection neuron
Kim et al., 2013; Blackwell et al., 2019
cerebellar Purkinje cells Kotaleski et al., 2002; Gallimore et al., 2018
PKC → GluR2 S880pCerebellar Purkinje cells Gallimore et al., 2018
Table 2
List of LTP/LTD experiments in the cortex.

The first column labels the experimental data set and names the underlying study. The second column shows the considered synaptic pathway and the third column shows whether the observed LTP/LTD had a pre- or post-synaptic origin. The fourth and fifth columns show the frequency (in Hz) of stimulation and the number of pulses delivered, respectively: 10 × 4 means that 10 trains of 4 pulses with 10 ms interval (100 Hz) were delivered, and likewise, 25 × 5 means that 25 trains of 5 pulses with 10 ms interval were delivered. The sixth column tells whether the data were obtained in control conditions or under additional blockers or agonists. The seventh, eighth, ninth, and tenth columns show the relative change in synaptic strength 10, 15, and 20 min after the start of the stimulus protocol and an average SD of the relative synaptic strengths — these values were approximated from the LTP/LTD curves plotted in the underlying references. The rows correspond to experiments from a given reference that are divided to 11 different experimental data sets. Within each data set, the underlying system is assumed to be otherwise similar to the control except for the applied modifier: as an example, the chemical or genetic blockade of CaMKII activity (as performed in Ma et al., 2008 and Hardingham et al., 2003) is here expected to only affect the ability of CaMKII to autophosphorylate, and the rest of the model parameters are kept fixed. The experiments printed in grey were included in the underlying study, but were excluded from the main analyses of the present work (see main text). EC – entorhinal cortex; PFC – prefrontal cortex; BC – barrel cortex; ACC – anterior cingulate cortex; VC – visual cortex; AuC – auditory cortex; CC – corpus callosum. (*): The LFS of 900 3-ms pulses at 5 Hz in data sets VC-1 and VC-2 was replaced by 180 15-ms pulses at 1 Hz to decrease computational load in the optimisation.

Data setReferencePathwayPre/postFreq.NpulsesExperiment10 min15 min20 minSD
EC-1Ma et al., 2008horizontalmostly100100control1.31.41.30.1
postCaMKII blocked1.051.020.950.07
without post-syn. Ca2+1.051.051.10.09
EC-2Ma et al., 2008ascendingmostly100100control1.61.61.60.11
postPKA blocked1.41.41.40.13
without post-syn. Ca2+1.31.41.40.13
PFC-1Sáez-Briones et al., 2015CC→PFCn/a312156control2.01.981.90.08
without β-adrenergic ligand1.341.41.360.09
PFC-2Flores et al., 2011CC→PFCn/a312156control1.71.61.640.12
without β−1-receptor agonist1.431.451.430.1
BCHardingham et al., 2003L4→L2/3n/a510 × 4control1.351.41.30.09
CaMKII mutant1.251.21.10.09
ACCSong et al., 2017L5/6 → L2/3post510 × 4control1.551.41.40.05
without s8451.11.051.050.07
without s8311.351.41.30.1
PFC-3Zhou et al., 2013L2/3 → L2/3mostly0.150control1.31.41.40.14
postwithout β−1-receptor agonist1.11.21.20.13
VC-1Kirkwood et al., 1997L4 → L3n/a510 × 4(CTR, HFS)1.31.261.260.07
(adult)(without CaMKII, HFS)1.021.021.020.02
5900*(CTR, LFS)n/a0.950.950.05
(without CaMKII, LFS)n/a0.880.930.03
VC-2Kirkwood et al., 1997L4 → L3n/a510 × 4(CTR, HFS)1.21.181.180.05
(4–5 w)(without CaMKII, HFS)1.071.091.080.03
5900*(CTR, LFS)n/a0.790.820.03
(without CaMKII, LFS)n/a0.820.890.03
AuC-1Kotak et al., 2007L6 → L5n/a125 × 5LTP-expressing cells1.981.581.930.19
AuC-2Kotak et al., 2007L6 → L5n/a125 × 5LTD-expressing cells0.770.680.670.09
Table 3
List of model reactions.

(A) The reaction-rate units are in 1/ms, 1/(nMms), 1/(nM2ms), 1/(nM3ms), or 1/(nM4ms), depending on the number of reactants. Reactions are grouped by similar modes of action and identical forward and backward rates. The denominators 𝐗, 𝐘, and 𝐙 represent groups of species detailed below. †: backward reaction rate proportional to [PKAc], not to [PKAc]2. (B) Groups of species as used in panel (A).

(A)Forw.Backw.Forw.Backw.
IDReactionRateRateIDReactionRateRate
1Ca + PMCA ⇌ PMCACa5e-050.00771GluR1𝐗22 + 𝐘22 ⇌ GluR1𝐙222.78e-080.002
2PMCACa ⇌ PMCA + CaOut0.00350.072GluR1𝐗23 ⇌ GluR1 𝐘23𝐙230.00050
3Ca + NCX ⇌ NCXCa1.68e-050.011273GluR1𝐗24 + PKAc ⇌ GluR1𝐙244e-060.024
4NCXCa ⇌ NCX + CaOut0.00560.074GluR1𝐗25 + PP1 ⇌ GluR1𝐙258.7e-070.00068
5CaOut + Leak ⇌ CaOutLeak1.5e-060.001175GluR1𝐗26 ⇌ GluR1 𝐘26 + PP10.000170
6CaOutLeak ⇌ Ca + Leak0.00110.076GluR1𝐗27 + PP1 ⇌ GluR1𝐙278.75e-070.0014
7Ca + Calbin ⇌ CalbinC2.8e-050.019677GluR1𝐗28 ⇌ GluR1 𝐘28 + PP10.000350
8L ⇌ LOut0.00052e-0978GluR1𝐗29 + PP2BCaMCa4 ⇌ GluR1𝐙292.01e-060.008
9L + R ⇌ LR5.555e-060.00579GluR1𝐗30 ⇌ GluR1𝐘30 + PP2BCaMCa40.0020
10LR + Gs ⇌ LRGs6e-071e-0680GluR1𝐗31 ⇌ GluR1_memb𝐗312e-078e-07
11Gs + R ⇌ GsR4e-083e-0781GluR1_S845𝐗32
12GsR + L ⇌ LRGs2.5e-060.0005⇌ GluR1_memb_S845𝐗323.28e-058e-06
13LRGs ⇌ LRGsbg + GsaGTP0.020.082PDE1 + CaMCa4 ⇌ PDE1CaMCa40.00010.001
14LRGsbg ⇌ LR + Gsbg0.080.083PDE1CaMCa4 + cAMP ⇌ PDE1CaMCa4cAMP4.6e-060.044
15𝐗1 + PKAc ⇌ PKAc𝐗18e-070.0044884PDE1CaMCa4cAMP ⇌ PDE1CaMCa4 + AMP0.0110.0
16PKAc𝐗2 ⇌ p𝐗2 + PKAc0.0010.085AMP ⇌ ATP0.0010.0
17ppLR + PKAc ⇌ PKAcppLR1.712e-050.0044886PDE4 + cAMP ⇌ PDE4cAMP2.166e-050.0034656
18pppLR + PKAc ⇌ PKAcpppLR0.0017120.0044887PDE4cAMP ⇌ PDE4 + AMP0.0172330.0
19ppppLR + Gi ⇌ ppppLRGi0.000150.0002588𝐗33 + 𝐘33 ⇌ PKAc 𝐙332.5e-078e-05
20ppppLRGi ⇌ ppppLRGibg + GiaGTP0.0001250.089PKAc𝐗34 ⇌ pPDE4𝐘34 + PKAc2e-050.0
21pppp𝐗3 ⇌ pppp𝐘3 + Gibg0.0010.090pPDE4 ⇌ PDE42.5e-060.0
22𝐗4𝐗42.5e-060.091pPDE4 + cAMP ⇌ pPDE4cAMP0.0004331750.069308
23pp𝐗5 ⇌ p𝐗52.5e-060.092pPDE4cAMP ⇌ pPDE4 + AMP0.34466740.0
24R + PKAc ⇌ PKAcR4e-080.0044893PKAcAMP4 ⇌ PKAr + 2*PKAc0.000242.55e-05
25pR + PKAc ⇌ PKAcpR4e-070.0044894Ca + fixedbuffer ⇌ fixedbufferCa0.000420.0
26ppR + PKAc ⇌ PKAcppR4e-060.0044895Glu ⇌ GluOut0.00052e-10
27pppR + PKAc ⇌ PKAcpppR0.00040.0044896Ca + PLC ⇌ PLCCa4e-070.001
28ppppR + Gi ⇌ ppppRGi7.5e-050.00012597GqaGTP + PLC ⇌ PLCGqaGTP7e-070.0007
29ppppRGi ⇌ ppppRGibg + GiaGTP6.25e-050.098Ca + PLCGqaGTP ⇌ PLCCaGqaGTP8e-050.04
30GsaGTP ⇌ GsaGDP0.010.099GqaGTP + PLCCa ⇌ PLCCaGqaGTP0.00010.01
31GsaGDP + Gsbg ⇌ Gs0.10.0100PLCCa + Pip2 ⇌ PLCCaPip23e-080.01
32GiaGTP ⇌ GiaGDP0.0001250.0101PLCCaPip2 ⇌ PLCCaDAG + Ip30.00030.0
33GiaGDP + Gibg ⇌ Gi0.001250.0102PLCCaDAG ⇌ PLCCa + DAG0.20.0
34GsaGTP + AC1 ⇌ AC1GsaGTP3.85e-050.01103PLCCaGqaGTP + Pip2 ⇌ PLCCaGqaGTPPip21.5e-050.075
35AC1 𝐗6 + CaMCa4 ⇌ AC1 𝐙66e-060.0009104PLCCaGqaGTPPip2 ⇌ PLCCaGqaGTPDAG + Ip30.250.0
36𝐗7 + ATP ⇌ 𝐙71e-052.273105PLCCaGqaGTPDAG ⇌ PLCCaGqaGTP + DAG1.00.0
37AC1GsaGTPCaMCa4ATP106Ip3degrad + PIkinase ⇌ Ip3degPIk2e-060.001
⇌ cAMP + AC1GsaGTPCaMCa40.028420.0107Ip3degPIk ⇌ PIkinase + Pip20.0010.0
38𝐗8 + 𝐘8 ⇌ AC1Gsa𝐙86.25e-050.01108PLC𝐗35 ⇌ PLC𝐘35 + GqaGDP0.0120.0
39𝐗9 ⇌ cAMP + 𝐙90.0028420.0109GqaGTP ⇌ GqaGDP0.0010.0
40AC1GiaGTPCaMCa4ATP110GqaGDP ⇌ Gqabg0.010.0
⇌ cAMP + AC1GiaGTPCaMCa40.00056840.0111Ca + DGL ⇌ CaDGL0.0001250.05
41AC1CaMCa4ATP ⇌ cAMP + AC1CaMCa40.0056840.0112DAG + CaDGL ⇌ DAGCaDGL5e-070.001
42AC8 + CaMCa4 ⇌ AC8CaMCa41.25e-060.001113DAGCaDGL ⇌ CaDGL + 2AG0.000250.0
43CaM + 2*Ca ⇌ CaMCa21.7e-080.035114Ip3 ⇌ Ip3degrad0.010.0
44𝐗10 + Ca ⇌ 𝐙101.4e-050.2281152AG ⇌ 2AGdegrad0.0050.0
45𝐗11 + Ca ⇌ 𝐙112.6e-050.064116DAG + DAGK ⇌ DAGKdag7e-080.0008
46CaM + Ng ⇌ NgCaM2.8e-050.036117DAGKdag ⇌ DAGK + PA0.00020.0
47CaM + PP2B ⇌ PP2BCaM4.6e-061.2e-06118Ca + PKC ⇌ PKCCa1.33e-050.05
48CaMCa𝐗12 + PP2B ⇌ PP2B𝐙124.6e-051.2e-06119PKCCa + DAG ⇌ PKCt1.5e-080.00015
49PP2BCaM + 2*Ca ⇌ PP2BCaMCa21.7e-070.35120Glu + MGluR ⇌ MGluR_Glu1.68e-080.0001
50CaMCa4 + CK ⇌ CKCaMCa41e-050.003121MGluR_Glu ⇌ MGluR_Glu_desens6.25e-051e-06
512*CKCaMCa4 ⇌ Complex1e-070.01122Gqabg + MGluR_Glu ⇌ MGluR_Gqabg_Glu9e-060.00136
52CKpCaMCa4 + CKCaMCa4 ⇌ pComplex1e-070.01123MGluR_Gqabg_Glu ⇌ GqaGTP + MGluR_Glu0.00150.0
53CK𝐗13 + Complex ⇌ CK𝐗13 + pComplex1e-070.0124GluR2𝐗36 + PKC𝐘36 ⇌ GluR2𝐙364e-070.0008
542*Complex ⇌ Complex + pComplex1e-050.0125GluR2𝐗37 ⇌ GluR2𝐘37 + PKC𝐙370.00470
55Complex + pComplex ⇌ 2*pComplex3e-050.0126GluR2𝐗38 + PP2A ⇌ GluR2𝐙385e-070.005
56CKpCaMCa4 ⇌ CaMCa4 + CKp8e-071e-05127GluR2𝐗39 ⇌ GluR2𝐘39 + PP2A0.000150
57CKp𝐗14 + PP1 ⇌ CKp𝐙144e-090.00034128GluR2𝐗40 ⇌ GluR2_memb𝐗400.000245450.0003
58CKp𝐗15 ⇌ PP1 + CK𝐙158.6e-050.0129GluR2_S880𝐗41 ⇌ GluR2_memb_S880𝐗410.00550.07
59PKA + 4*cAMP ⇌ PKAcAMP41.6e-156e-05130ACh + M1R ⇌ AChM1R9.5e-080.0025
60Epac1 + cAMP ⇌ Epac1cAMP3.1e-086.51e-05131Gqabg + AChM1R ⇌ AChM1RGq2.4e-050.00042
61I1 + PKAc ⇌ I1PKAc1.4e-060.0056132Gqabg + M1R ⇌ M1RGq5.76e-070.00042
62I1PKAc ⇌ Ip35 + PKAc0.00140.0133ACh + M1RGq ⇌ AChM1RGq3.96e-060.0025
63Ip35 + PP1 ⇌ Ip35PP11e-061.1e-06134AChM1RGq ⇌ GqaGTP + AChM1R0.00050.0
64Ip35𝐗16 + PP2BCaMCa4 ⇌ Ip35PP2B𝐙169.625e-050.33135ACh ⇌0.0060
65Ip35PP2B𝐗17 ⇌ I1 + PP2B𝐗170.0550.0136Ca + PLA2 ⇌ CaPLA26e-070.003
66PP1PP2BCaMCa4 ⇌ PP1 + PP2BCaMCa40.00150.0137CaPLA2 + Pip2 ⇌ CaPLA2Pip22.2e-050.444
67GluR1𝐗18 + PKAc ⇌ GluR1𝐙184.02e-060.024138CaPLA2Pip2 ⇌ CaPLA2 + AA0.1110.0
68GluR1𝐗19 ⇌ GluR1𝐘19 + PKAc0.0060139AA ⇌ Pip20.0010.0
69GluR1𝐗20 + CK𝐘20 ⇌ GluR1𝐙202.224e-080.0016140PKCt + AA ⇌ PKCp5e-091.76e-07
70GluR1𝐗21 ⇌ GluR1𝐘21 + CK𝐙210.00040
(B)
𝐗1 ∈ {LR, pLR}(𝐗23, 𝐘23, 𝐙23) ∈ { (_CKpCam, _S831, CKpCaMCa4), (_PKCt,
𝐗2 ∈ {LR, pLR, ppLR, pppLR, R, pR, ppR, pppR}_S831, PKCt), (_PKCp, _S831, PKCp), (_S845_CKpCam, _S845_S831,
(𝐗3, 𝐘3) ∈ { (LRGibg, LR), (RGibg, R) }CKpCaMCa4), (_S845_PKCt, _S845_S831, PKCt), (_S845_PKCp,
𝐗4 ∈ {LR, R, pR}_S845_S831, PKCp), (_memb_CKpCam, _memb_S831, CKpCaMCa4),
𝐗5 ∈ {LR, pLR, ppLR, pR, ppR}(_memb_PKCt, _memb_S831, PKCt), (_memb_PKCp, _memb_S831, PKCp),
(𝐗6, 𝐙6) ∈ { (GsaGTP, GsaGTPCaMCa4), (GsaGTPGiaGTP,(_memb_S845_CKpCam, _memb_S845_S831, CKpCaMCa4),
GsaGTPGiaGTPCaMCa4), ({}, CaMCa4) }(_memb_S845_PKCt, _memb_S845_S831, PKCt), (_memb_S845_PKCp,
(𝐗7, 𝐙7) ∈ { (AC1GsaGTPCaMCa4, AC1GsaGTPCaMCa4ATP),_memb_S845_S831, PKCp) }
(AC1GsaGTPGiaGTPCaMCa4, AC1GsGiCaMCa4ATP), (AC1GiaGTPCaMCa4,(𝐗24, 𝐙24) ∈ { (_S831, _S831_PKAc), (_memb_S831,
AC1GiaGTPCaMCa4ATP), (AC1CaMCa4, AC1CaMCa4ATP), (AC8CaMCa4,_memb_S831_PKAc) }
AC8CaMCa4ATP) }(𝐗25, 𝐙25) ∈ { (_S845, _S845_PP1), (_memb_S845,
(𝐗8, 𝐘8, 𝐙8) ∈ { (GiaGTP, AC1GsaGTP, GTPGiaGTP), (GiaGTP,_memb_S845_PP1) }
AC1CaMCa4, GTPCaMCa4), (AC1GiaGTP, GsaGTP, GTPGiaGTP) }(𝐗26, 𝐘26) ∈ { (_S845_PP1, {}), (_memb_S845_PP1,
(𝐗9, 𝐙9) ∈ { (AC1GsGiCaMCa4ATP, AC1GsaGTPGiaGTPCaMCa4),_memb) }
(AC8CaMCa4ATP, AC8CaMCa4) }(𝐗27, 𝐙27) ∈ { (_S845_S831, _S845_S831_PP1), (_S831,
(𝐗10, 𝐙10) ∈ { (CaMCa2, CaMCa3), (PP2BCaMCa2,_S831_PP1), (_memb_S845_S831, _memb_S845_S831_PP1),
PP2BCaMCa3) }(_memb_S831, _memb_S831_PP1) }
(𝐗11, 𝐙11) ∈ { (CaMCa3, CaMCa4), (PP2BCaMCa3,(𝐗28, 𝐘28) ∈ { (_S845_S831_PP1, _S845), (_S845_S831_PP1,
PP2BCaMCa4) }_S831), (_S831_PP1, {}), (_memb_S845_S831_PP1, _memb_S845),
(𝐗12, 𝐙12) ∈ { (2, CaMCa2), (4, CaMCa4) }(_memb_S845_S831_PP1, _memb_S831), (_memb_S831_PP1,
𝐗13 ∈ {pCaMCa4, CaMCa4}_memb) }
(𝐗14, 𝐙14) ∈ { ({}, PP1), (CaMCa4, CaMCa4PP1) }(𝐗29, 𝐙29) ∈ { (_S845, _S845_PP2B), (_S845_S831,
(𝐗15, 𝐙15) ∈ { (PP1, {}), (CaMCa4PP1, CaMCa4) }_S845_S831_PP2B), (_memb_S845, _memb_S845_PP2B),
(𝐗16, 𝐙16) ∈ { ({}, CaMCa4), (PP1, P2BCaMCa4) }(_memb_S845_S831, _memb_S845_S831_PP2B) }
𝐗17 ∈ {CaMCa4, P2BCaMCa4}(𝐗30, 𝐘30) ∈ { (_S845_PP2B, {}), (_S845_S831_PP2B,
(𝐗18, 𝐙18) ∈ { ({}, _PKAc), (_memb, _memb_PKAc) }_S831), (_memb_S845_PP2B, _memb), (_memb_S845_S831_PP2B,
(𝐗19, 𝐘19) ∈ { (_PKAc, _S845), (_S831_PKAc, _S845_S831),_memb_S831) }
(_memb_PKAc, _memb_S845), (_memb_S831_PKAc,𝐗31 ∈ {{}, _PKAc, _CKCam, _CKpCam, _CKp, _PKCt, _PKCp,
_memb_S845_S831) }_S831, _S831_PKAc, _S831_PP1}
(𝐗20, 𝐘20, 𝐙20) ∈ { ({}, CaMCa4, _CKCam), ({}, p,𝐗32 ∈ {{}, _CKCam, _CKpCam, _CKp, _PKCt, _PKCp, _S831,
_CKp), (_S845, CaMCa4, _S845_CKCam), (_S845, p, _S845_CKp),_PP1, _S831_PP1, _PP2B, _S831_PP2B}
(_memb, CaMCa4, _memb_CKCam), (_memb, p, _memb_CKp),(𝐗33, 𝐘33, 𝐙33) ∈ { (PKAc, PDE4, PDE4), (PDE4cAMP, PKAc,
(_memb_S845, CaMCa4, _memb_S845_CKCam), (_memb_S845, p,_PDE4_cAMP) }
_memb_S845_CKp) }(𝐗34, 𝐘34) ∈ { (PDE4, {}), (_PDE4_cAMP, cAMP) }
(𝐗21, 𝐘21, 𝐙21) ∈ { (_CKCam, _S831, CaMCa4), (_CKp, _S831,(𝐗35, 𝐘35) ∈ { (GqaGTP, {}), (CaGqaGTP, Ca) }
p), (_S845_CKCam, _S845_S831, CaMCa4), (_S845_CKp, _S845_S831,(𝐗36, 𝐘36, 𝐙36) ∈ { ({}, t, _PKCt), ({}, p, _PKCp),
p), (_memb_CKCam, _memb_S831, CaMCa4), (_memb_CKp, _memb_S831,(_memb, t, _memb_PKCt), (_memb, p, _memb_PKCp) }
p), (_memb_S845_CKCam, _memb_S845_S831, CaMCa4),(𝐗37, 𝐘37, 𝐙37) ∈ { (_PKCt, _S880, t), (_PKCp, _S880, p),
(_memb_S845_CKp, _memb_S845_S831, p) }(_memb_PKCt, _memb_S880, t), (_memb_PKCp, _memb_S880, p) }
(𝐗22, 𝐘22, 𝐙22) ∈ { ({}, CKpCaMCa4, _CKpCam), ({},(𝐗38, 𝐙38) ∈ { (_S880, _S880_PP2A), (_memb_S880,
PKCt, _PKCt), ({}, PKCp, _PKCp), (_S845, CKpCaMCa4,_memb_S880_PP2A) }
_S845_CKpCam), (_S845, PKCt, _S845_PKCt), (_S845, PKCp,(𝐗39, 𝐘39) ∈ { (_S880_PP2A, {}), (_memb_S880_PP2A,
_S845_PKCp), (_memb, CKpCaMCa4, _memb_CKpCam), (_memb, PKCt,_memb) }
_memb_PKCt), (_memb, PKCp, _memb_PKCp), (_memb_S845, CKpCaMCa4,𝐗40 ∈ {{}, _PKCt, _PKCp}
_memb_S845_CKpCam), (_memb_S845, PKCt, _memb_S845_PKCt),𝐗41 ∈ {{}, _PP2A}
(_memb_S845, PKCp, _memb_S845_PKCp) }
Table 4
List of initial concentrations of molecular species.

All non-mentioned species have an initial concentration of 0 nM.

SpeciesConc. (nM)SpeciesConc. (nM)SpeciesConc. (nM)
CaOutextracell. Ca2+1900000AMPadenosine monophosphate980Pip2phosphatidylinositol 4,5-bisphosphate24000
Leakleak channels2000Ngneurogranin20000PIkinasephosphatidylinositol kinase290
Calbincalbindin150000CaMcalmodulin60000Ip3degPIkIp3-bound PI kinase400
CalbinCCa2+-bound calbindin15000PP2Bprotein phosphatase 2B2300PKCprotein kinase C15000
LOutextracell. β-adr. ligand2500000CKCaMKII23000DAGdiacylglycerol90
Epac1Epac1500PKAprotein kinase A6400DAGKDAG kinase300
PMCACa2+ pump22000I1inhibitor-12200DGLDAG lipase1600
NCXCa2+ exchanger540000PP1protein phosphatase 11600CaDGLCa2+-bound DAG lipase250
Lβ-adrenergic ligand10GluR1AMPAR subunit type 1180DAGCaDGLCa2+-and DAG-bound DAG lipase90
Rβ-adrenergic receptor1600GluR1_membmembrane-inserted GluR190Ip3degraddegraded Ip3600
GsS-type G-protein13000PDE4phosphodiesterase type 4670GluR2AMPAR subunit type 214
GiI-type G-protein2600fixedbufferimmobile buffer500000GluR2_membmembrane-inserted GluR2256
AC1adenylyl cyclase type 1430.0mGluRmetab. glutamate receptor800PP2Aprotein phosphatase 2A500
ATPadenosine triphosphate2000000GluOutextracell. glutamate1000000M1Racetylcholine receptor M1450
AC8adenylyl cyclase type 8370GqabgQ-type G-protein1400PLA2phospholipase A21000
PDE1phosphodiesterase type 112000PLCphospholipase C250

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