Inhibitory control of site-specific synaptic plasticity in a model CA1 pyramidal neuron

BioSystems 130 (2015) 37–50

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Inhibitory control of site-specific synaptic plasticity in a model CA1 pyramidal neuron Aušra Saudargiene_ a, * , Bruce P. Graham b a b

Department of Informatics, Vytautas Magnus University, Kaunas LT-44404, Lithuania Computer Science and Mathematics, School of Natural Sciences, University of Stirling, Stirling FK9 4LA, UK

A R T I C L E I N F O

A B S T R A C T

Article history: Received 5 March 2014 Received in revised form 31 October 2014 Accepted 6 March 2015 Available online 10 March 2015

A computational model of a biochemical network underlying synaptic plasticity is combined with simulated on-going electrical activity in a model of a hippocampal pyramidal neuron to study the impact of synapse location and inhibition on synaptic plasticity. The simulated pyramidal neuron is activated by the realistic stimulation protocol of causal and anticausal spike pairings of presynaptic and postsynaptic action potentials in the presence and absence of spatially targeted inhibition provided by basket, bistratified and oriens-lacunosum moleculare (OLM) interneurons. The resulting Spike-timingdependent plasticity (STDP) curves depend strongly on the number of pairing repetitions, the synapse location and the timing and strength of inhibition. ã 2015 Elsevier Ireland Ltd. All rights reserved.

Keywords: Spike-timing-dependent plasticity CA1 pyramidal neuron Dendritic spike Inhibitory interneurons

1. Introduction Synaptic plasticity is believed to underly the biological basis of learning and memory. Bidirectional long-term synaptic changes are induced by coincident activation of presynaptic and postsynaptic neurons (Markram et al., 1997; Bi and Poo, 1998, 2001). If the presynaptic spike precedes the postsynaptic spike within a short time window, the synapse undergoes long-term potentiation (LTP), and it exhibits long-term depression (LTD) if the temporal order is reversed. Variations on this classic picture of spike timing dependent plasticity (STDP) have been found in the hippocampus, with the activity windows for LTP and LTD depending on the frequency of input–output spike pairing, the duration of such pairing and spike bursting in the postsynaptic cell (Nishiyama et al., 2000; Wittenberg and Wang, 2006; Buchanan and Mellor, 2010). Physiological observations suggest that the form of plasticity at a synapse depends not only on the timing of the presynaptic and postsynaptic activity, but also on the location of the synapse on the dendritic tree (Golding et al., 2002; Häusser and Mel, 2003; Froemke et al., 2005, 2010; Letzkus et al., 2006; Sjöström et al., 2006, 2008). Strong postsynaptic depolarization, necessary for induction of synaptic modifications, is provided by somatic back-propagating spikes in dendritic regions proximal to the soma (Bi and Poo, 2001; Wittenberg and Wang, 2006). In distal apical dendrites, local

* Corresponding author. Tel.: +370 37 327900; fax: +370 37 327896. _ E-mail address: [email protected] (A. Saudargiene). http://dx.doi.org/10.1016/j.biosystems.2015.03.001 0303-2647/ ã 2015 Elsevier Ireland Ltd. All rights reserved.

dendritic regenerative spikes might drive synaptic modifications (Magee and Johnston, 1997; Golding et al., 2002; Froemke et al., 2005) even in the absence of somatic spiking. Thus, synapses undergo changes according to local, rather than global, synaptic plasticity rules (Letzkus et al., 2006; Sjöström and Häusser, 2006). In addition, hippocampal CA1 pyramidal neurons receive inhibitory inputs from different classes of inhibitory interneurons (Klausberger et al., 2003, 2004). It was shown experimentally that inhibition of the proximal apical dendrites is responsible for the switch between the symmetrical and asymmetrical STDP rules at the synapses of the CA1 pyramidal neuron (Tsukada et al., 2005). However, how the dendritic location and inhibitory inputs shape the local plasticity rules at a synapse under realistic stimulation protocols remains unclear. Numerous modelling studies investigate mechanisms of synaptic plasticity, but only a limited number of studies address these further issues (Bar-Ilan et al., 2013; Cutsuridis, 2011, 2012). Changes in STDP shape under the influence of dendritic inhibition, triggered at various frequencies, strengths and relative timings with the presynaptic and postsynaptic activity, were analyzed in Cutsuridis (2011, 2012),). The results showed that dendritic inhibition can induce LTD for causal presynaptic and postsynaptic action potential (AP) pairings. The CA1 pyramidal neuron model was simplified, consisting of two compartments, which limited the spatial effects that could be explored, and only GABA-A dendritic inhibition was implemented. Induction and maintenance of synaptic modifications involve complex biochemical pathways (Bhalla and Iyengar, 1999; Lisman et al., 1997, 2002; Malenka and Bear, 2004). The trigger of the

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protein signalling cascade reactions is the intracellular calcium mediated by NMDAr–gated channels and voltage-sensitive calcium channels (Koester and Sakmann, 1998; Yuste et al. 1999). Synaptic NMDAr activation is necessary for synaptic plasticity induction (Magee and Johnson, 1997). Brief, high calcium increase leads to potentiation, and prolonged moderate calcium levels induce depression (Mizuno et al., 2001). Intracellular calcium activates CaMKII and protein phosphatases PP2A, PP2B. CaMKII phosphorylates and phosphatases dephosphorylate AMPA-type glutamate receptors, and these changes in AMPAR phosphorylation state are thought to underlie LTP and LTD (Lisman et al., 1997). Here we use a detailed compartmental model of a CA1 pyramidal neuron (Poirazi et al., 2003), embedded in a CA1 microcircuit model (Cutsuridis et al., 2010), to explore the influence of dendritic synapse location and spatially targetted inhibition on synaptic plasticity at the synapses formed on the stratum radiatum (SR) and stratum lacunosum moleculare (SLM) dendrites of this neuron. Simulation of multiple causal and anticausal pairings of presynaptic action potentials and postsynaptic activity are performed to model calcium entry through NMDA receptors and voltage-gated calcium channels in the SR and SLM spines. The obtained calcium signals are used in the STDP model, derived from models by Graupner and Brunel (2007) and Pi and Lisman (2008). The model simulates the activity of calcium– calmodulin dependent protein kinase II (CaMKII) and protein phosphatase 2 (PP2A), leading to changes in AMPA receptors and thus LTP or LTD. We employ a detailed, calcium-based model of synaptic plasticity to capture the synaptic modification dependence not only on the calcium amplitude, but the overall time course of calcium signal in a spine of the CA1 pyramidal neuron. The modeled synaptic modifications show sensitivity not only to the timing of the pre- and postsynaptic activity, but also to the frequency and duration of simulation. The model makes use of the well-established existing models with explicit LTP (Graupner and Brunel, 2007) and LTD (Pi and Lisman, 2008) pathways that can be tuned to match a wide variety of experimental data (Wittenberg and Wang, 2006).

2. Methodology 2.1. Model of CA1 pyramidal neuron microcircuit A detailed multicompartmental model of a CA1 pyramidal cell (Poirazi et al., 2003) embedded in a model of the CA1 pyramidal neuron microcircuit (Cutsuridis et al., 2010) (Fig. 1) is used to simulate spine head calcium signals under realisitic activity conditions, as calcium is a critical trigger of synaptic modifications. Full details of the cell model can be found in Poirazi et al. (2003). Spines with AMPA and NMDAr-gated channels are formed on proximal, medial and distal SR apical dendrites (94 mm, 121 mm, 157 mm from the soma) and on the SLM apical dendrite (413 mm from the soma) of the CA1 pyramidal cell. Spine head diameter and length equal 0.5 mm, spine neck diameter is 0.2 mm and length is 1 mm. Spines contain the same ion channels as their parent dendrites. However, calcium channels were not inserted in the SLM spine as there is a lack of knowledge of the calcium levels observed in spines on distal apical dendrites during dendritic spike induction. A kinetic model of the NMDAr-gated channel is used (VargasCaballero and Robinson, 2004), and the peak synaptic conductance is adjusted to obtain NMDAr-mediated calcium level changes of 13 mM and 0.5 mM in spines at 0 mV and –70 mV, respectively, following a single presynaptic action potential (Sabatini et al., 2002). The AMPAr-mediated synaptic response is modeled as a double exponential function:

Fig. 1. Microcircuit of a CA1 pyramidal neuron. Compartmental model of a CA1 pyramidal neuron (Poirazi et al., 2003) with the added AMPA/NMDA synapses on SLM spine and SR spines, embedded in a microcircuit consisting of OLM, bistratified (BSC) and basket (BC) inhibitory interneurons (Cutsuridis et al., 2010). Synaptic inputs come from CA3 region and entorhinal cortex (EC). CA1 Stratum pyramidale (SP), SR and SLM layers are shown schematically.

gsyn ¼ gsyn ðeðttpre Þ=tfall  eðttpre Þ=trise Þ;

(1)

where t rise is the rising time constant, t fall is the decay time constant, gsyn is a peak synaptic conductance leading to a local EPSP of 3 mV, tpre is the time of the synapse activation. Time constants, peak synaptic conductance and reversal potential of the AMPA synapse are presented in Tables 1 and 2. In addition, the CA1 pyramidal neuron has inhibitory GABA-A and GABA-B synapses to capture the influence of the basket, bistratified and OLM inhibitory interneurons (Cutsuridis et al., 2010). Model basket, bistratified and OLM cells that trigger the activity of these inhibitory synapses are adopted from Cutsuridis et al. (2010). Six inhibitory GABA-A and six inhibitory GABA-B synapses are placed 45 mm, 52 mm, 70 mm, 156 mm, 210 mm, 233 mm from the soma to provide the effective inhibition of the SR dendritic region. These synapses are activated by a bistratified interneuron, which receives the same excitatory input from the CA3 region as a SR synapse. One GABA-A synapse is located on the soma of the CA1 pyramidal neuron, and its activity, triggered by a basket interneuron, prevents CA1 pyramidal neuron from action potential generation. Two GABA-A and two GABA-B synapses in the SLM dendritic region (413 mm and 414 mm from the soma, left and right distal apical dendrites) are sufficient to inhibit distal dendrites and are activated by the OLM interneuron. Basket and OLM interneurons are recurrently excited by somatic action potentials from the CA1 pyramidal cell. GABA-A and GABA-B synapses are implemented using a double exponential function Eq. (1) with the synaptic parameters as in

Table 1 Time constants, reversal potential and peak synaptic conductance of AMPA, GABAA, GABA-B synapses and two somatic and one SLM dendritic excitatory synapses used to induce doublets of somatic action potentials and a dendritic spike in the CA1 pyramidal neuron model. Synapse

Rising time constant Decay time constant t fall, ms t rise, ms

Reversal potential, mV

AMPA GABA-A GABA-B Excitatory somatic slow Excitatory somatic fast Excitatory SLM dendritic

0.5 1 35 4

3 8 100 5

0 75 75 0

0.2

2

0

0.2

0.5

0

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Table 2 Peak synaptic conductances (nS for AMPA, GABA-A, GABA-B, nS/cm2 for NMDA). Type of postsynaptic receptor is indicated in parentheses. Presynaptic

EC CA3

BC BSC

OLM CA1 pyramidal neuron PC – – –

Postsynaptic CA1 pyramidal neuron PC 2.2 (AMPA) 65 (NMDA) 2.2 (AMPA) 13 (strong AMPA) 65 (NMDA) 500 (GABA-A) 10 (strong GABA-A) 1 (strong GABA-B) 5 (weak GABA-A) 0.5 (weak GABA-B) 50 (GABA-A) 10 (GABA-B) – 20 (excitatory somatic slow) 20 (excitatory somatic fast) 500 (excitatory SLM dendritic)

BC

BSC

OLM

–

–

–

–

5 – (AMPA)

– –

– –

– –

–

–

–

100 (AMPA) –

– –

10 (AMPA) –

–

–

–

–

–

–

Fig. 2. Protein signaling cascades governing AMPAr changes ( denotes inhibition).

Cutsuridis et al. (2010). All parameters of the synapses are listed in Tables 1 and 2. Calcium concentration in a spine is modelled as (Badoual et al., 2006): d ICa ð½Ca2þ 0  ½Ca2þ Þ ½Ca2þ  ¼  þ ; dt 2Fd  18 t Ca

(2)

where [Ca2+] is the calcium concentration in a spine, [Ca2 ]0 = 0.1 mM is the calcium concentration in the spine at rest, ICa is the calcium current, F is the Faraday constant, d = 0.1 mm is the depth of dendritic shell, t Ca = 15 ms is the time constant of calcium concentration decay (Badoual et al., 2006). Factor 18 reflects the influence of the endogenous buffers (Badoual et al., 2006). Calcium transients are obtained by two different stimulation protocols applied to induce synaptic modifications at SR and SLM synapses: (1) A presynaptic action potential at a SR spine close to the soma is paired with a burst-like doublet of somatic action potentials or a single action potential, following the experimental protocol of Wittenberg and Wang (2006); postsynaptic action potentials in doublets are spaced 10 ms apart. (2) A presynaptic action potential at a distal SLM spine is paired with a dendritic spike, as in Golding et al. (2002). Somatic action potentials and dendritic spikes are induced by post-synaptic depolarisation generated by two somatic and one dendritic artificial excitatory synapses, respectively, modeled with double exponential functions Eq. (1). Parameters of the artificial somatic and dendritic synapses are presented in Tables 1 and 2. Stimulation protocols are applied 30 and 100 times at 5 Hz and at 0.5 Hz, as per Wittenberg and Wang (2006).

+

2.2. Model of synaptic plasticity The model of synaptic plasticity is based on molecular pathways of LTP and LTD induction, involving activation of CaMKII (Graupner and Brunel, 2007) and phosphatase PP2A (Pi and Lisman, 2008), and the interaction of CaMKII and PP2A, that leads to the AMPA receptor phosphorylation or dephosphorylation (Fig. 2) underlying LTP and LTD (Lisman et al., 1997). These pathways are driven by the intracellular calcium signal in spines of the CA1 pyramidal neuron model. CaMKII is responsible for LTP induction: calcium binds to calmodulin and so activates CaMKII, resulting in AMPAr

denotes activation;

phosphorylation and synaptic strengthening. As CaMKII activity becomes high, the autophosphorylation process enables CaMKII to retain activity even in the absence of calcium-bound calmodulin. Protein phosphotase inhibitor-1 (I1) and type 1 protein phosphatase (PP1) chains control the rate of dephosphorylation of phosphorylated CaMKII subunits and enable a synapse to act like a binary switch. CaMKII dynamics is described by a complex bistable model consisting of a large set of differential equations (Graupner and Brunel, 2007). In the LTD induction pathway, phosphatase PP2A is dephosphorylated by calcium and being in its active form inhibits CaMKII, dephosphorylates AMPAr and leads to LTD. PP2A is bistable, and a high level of active PP2A triggers autodephosphorylation of PP2A and allows PP2A to stay activated for resting calcium concentrations. PP2A activity is described by a single differential equation (Pi and Lisman, 2008). In addition, CaMKII and PP2A mutually inhibit each other. If CaMKII wins over PP2A, AMPAr is potentiated, and it is depressed if PP2A activity overwhelms CaMKII. The inhibitory effect of PP2A on CaMKII is achieved by inserting the active PP2A in the dephosphorylation rate kD of phosphorylated CaMKII subunits in the model of Graupner and Brunel (2007): kD ¼

kPP1 PP1 þ kPP2A PP2A ; K M þ CaMKII

(3)

where kPP1 is the maximal dephosphorylation rate by PP1, PP1 is the concentration of active PP1, kPP2A is the maximal dephosphorylation rate by PP2A, PP2A is the concentration of active PP2A, KM is the Michaelis–Menten constant of CaMKII dephosphorylation (assumed to be identical for both PP1 and PP2A, for simplicity) and CaMKII is the concentration of phosphorylated CaMKII subunits. High PP2A activity leads to the increased dephosphorylation rate of phosphorylated CaMKII subunits, reduced levels of active CaMKII and LTD. The CaMKII-provided inhibition of PP2A is obtained by rewriting the PP2A equation (Pi and Lisman, 2008) and substituting the amount of phosphorylated CaMKII subunits in the second right-hand term responsible for PP2A inactivation: d PP2ATot  PP2A PP2A ¼ k11 PP2A dt K m11 þ PP2ATot  PP2A PP2A  k12 ðCaMKII þ CaMKII0 Þ þ k13 PP2A0 K m12 þ PP2A þ2 3 ½Ca  þ k14 3 ðPP2ATot  PP2AÞ; K m þ ½Caþ2 3

(4)

where PP2ATot is the total concentration of PP2A, CaMKII0 and PP2A0 are the basal concentration of phosphorylated CaMKII subunits and dephosphorylated PP2A, respectively, k11 is the rate constant of autodephosphorylation of PP2A, k12 is the rate constant of phosphorylation of PP2A by CaMKII, k13 is the rate constant of basal activity of PP2A, k14 is the rate constant of calciumdependent dephosphorylation of PP2A, Km11 and Km12 are the Michaelis–Menten constants for autodephosphorylation and

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phosphorylation of PP2A, and Km is a dissociation constant for calcium. High CaMKII activity results in reduced PP2A activity, AMPAr phosphorylation, and hence LTP. Active AMPAr, inserted into the postsynaptic membrane, is obtained as (Pi and Lisman, 2008): d AMPA ¼ kAMPA ðAMPATot  AMPAÞ  kAMPA AMPA; dt

(5)

where AMPA is the concentration of active, phosphorylated AMPAr; AMPATot is the total concentration of AMPA receptors, kAMPA and k-AMPA are the AMPAr phosphorylation and dephosphorylation rates, respectively. These latter rates are calculated as: kAMPA ¼ c1 CaMKII þ c2 ;

(6)

k AMPA ¼ c3 PP2A þ c4 ;

(7)

where c1 and c2 are scaling constants; c3 and c4 are rate constants independent of CaMKII and PP2A activities. These constants are calculated analytically to ensure phosphorylated AMPA rises up to 0.9 for highly phosphorylated CaMKII to reflect LTP; decreases to

0.1 for highly dephosphorylated PP2A to indicate LTD; and stays at 0.5 for basal AMPA level. Compared to Pi and Lisman (2008), all the parameters of the depression pathway were considerably decreased (reaction rates slowed) and tuned to guarantee the tristability of the system and matching to the experimental results (Wittenberg and Wang, 2006) taking the CaMKII model (Graupner and Brunel, 2007) as a reference. The full set of equations describing the plasticity model is presented in Appendix A. Parameter values are given in Table A1, Appendix A.

2.3. Stimulation protocols The following simulation protocols were applied to analyze the influence of dendritic synapse location on plasticity:  Single spike activation of synapses on the proximal, medial and

distal SR spines paired with a burst of two somatic action

Table A1 Parameters of the STDP model. Parameter

Value

Units

Definition and reference

CaMKII KPKA kPP2A kPP1 KM K1 K2 K3 K4 K5 k6

0.175 0.14 6000 0.4 0.1 0.025 0.32 0.4 0.1 6

mM

k7

6

s1

kPP1 k_PP1 nCaN

kPKA PP1Tot CaMKIITot CaMTot I1Tot

500 0.1 3 0.1 18 0.053 8 0.00359 100 0.2 33.3 0.1 1

(s mM)1 s1 – s1 s1 mM – s1 s1 mM mM mM mM

PKA half activity concentration; defines the LTP threshold; adjusted. Rate of CaMKII subunit dephosphorylation by PP2A; adjusted. Maximal rate of CaMKII subunit dephosphorylation by PP1 (Graupner and Brunel, 2007). Michaelis-Menten constant of CaMKII subunit dephosphorylation (Graupner and Brunel 2007). Dissociation constant to calcium binding to calmodulin (Graupner and Brunel, 2007). Dissociation constant to calcium binding to calmodulin (Graupner and Brunel, 2007). Dissociation constant to calcium binding to calmodulin (Graupner and Brunel, 2007). Dissociation constant to calcium binding to calmodulin (Graupner and Brunel, 2007). Dissociation constant between dephosphorylated CaMKII subunit and calmodulin (Graupner and Brunel, 2007). Rate of CaMKII subunit autophosphorylation; calmodulin is bound to the two interacting and not phosphorylated CaMKII subunits (Graupner and Brunel, 2007). Rate of CaMKII subunit autophosphorylation; calmodulin is bound to the phosphorylated CaMKII subunit and to the CaMKII subunit to be phosphorylated; or phosphorylated CaMKII subunit is calmodulin-free and the CaMKII subunit to be phosphorylated is bound with calmodulin (Graupner and Brunel, 2007). I1P, PP1 association rate (Graupner and Brunel, 2007). I1P, PP1 dissociation rate (Graupner and Brunel, 2007). Calcineurin Hill coefficient (Graupner and Brunel, 2007). Calcineurin base activity (Graupner and Brunel, 2007). Maximum calcium-calmodulin-dependent calcineurin activity (Graupner and Brunel, 2007). Calcineurin half activation concentration (Graupner and Brunel, 2007). PKA Hill coefficient (Graupner and Brunel, 2007). PKA base activity (Graupner and Brunel, 2007). Maximum calcium-calmodulin-dependent PKA activity (Graupner and Brunel, 2007). Total PP1 concentration (Graupner and Brunel, 2007). Total CaMKII concentration (Graupner and Brunel, 2007). Total calmodulin concentration (Graupner and Brunel, 2007). Total I1 concentration (Graupner and Brunel, 2007).

PP2A Km Km11 Km12 k11 k12 k13 k14 CaMKII0 PP2A0 PP2ATot

1.95 15 1 0.05 0.0026 0.025 2 71.4 0.0425 20

mM mM mM s1 s1 s1 s1 mM mM mM

Dissociation constant for calcium, defines the LTD threshold; adjusted. Michaelis-Menten constant for PP2A autodephosphorylation; adjusted. Michaelis-Menten constant for PP2A phosphorylation (Pi and Lisman, 2008). Rate constant of PP2A autodephosphorylation; adjusted. Rate constant of PP2A phosphorylation by CaMKII, adjusted. Rate constant of PP2A basal activity; adjusted. Rate constant of calcium-dependent PP2A dephosphorylation; adjusted. Basal concentration of phosphorylated CaMKII subunits; adjusted. Basal concentration dephosphorylated PP2A; adjusted. Total concentration PP2A (Pi and Lisman, 2008)

AMPAr c1 c2 c3 c4 AMPA

0.054 0.52 1.014 1 1

– – s1 s1 –

Scaling constant; adjusted. Scaling constant; adjusted. Rate constant independent from CaMKII activity; adjusted. Rate constant independent from PP2A activity; adjusted. Normalized total concentration of AMPAr (Pi and Lisman, 2008)

kCaN KCaN nPKA

s1 s1 mM mM mM mM mM mM s1

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potentials (spaced 10 ms apart), repeated 30 or 100 times at 5 Hz (following Wittenberg and Wang (2006)).  Single spike activation of a synapse on a SLM spine paired with a dendritic spike induced by a suprathreshold dendritic current injection, repeated 30 or 100 times at 5 Hz. The following simulation protocols of the pyramidal neuron were applied to analyze the influence of inhibitory interneurons:  Activation of a synapse on a proximal SR spine paired with a

doublet of postsynaptic action potentials at 5 Hz; soma of the pyramidal neuron is inhibited by a basket interneuron that is activated by the CA1 pyramidal cell somatic action potentials.  Activation of aproximal synapse on a SR spine paired with a doublet of postsynaptic action potentials at 5 Hz; medial apical dendrites of the pyramidal neuron are inhibited by a bistratified interneuron that is activated by the same CA3 inputs as the SR synapse.  Activation of a synapse on a SLM spine paired with a dendritic spike induced by a suprathreshold dendritic current injection at 5 Hz; distal apical dendrites of the pyramidal neuron are inhibited by an OLM neuron that is triggered by the CA1 pyramidal cell somatic action potentials induced by the dendritic spike. The numerical simulations were performed using NEURON (Hines and Carnevale, 1997) and Borland C++ running on a PC under Windows 7. The forward Euler’s integration method with a time step of 0.025 ms was used for the numerical integration. 3. Results Changes in strength at the synapses on SR and SLM spines of a CA1 pyramidal neuron are analyzed using the model of synaptic plasticity to uncover the influence of dendritic synapse location and spatially targeted inhibition, provided by basket, bistratified and OLM inhibitory interneurons. 3.1. Influence of dendritic synapse location 3.1.1. Plasticity rules of excitatory synapses in SR dendritic region To test the proposed model of synaptic plasticity, modifications at the synapse on the proximal SR spine, located 94 mm from the soma, are simulated and compared to physiological data (Wittenberg and Wang, 2006). A presynaptic spike is paired with a burst of two postsynaptic action potentials spaced 10 ms apart and the temporal difference T between the presynaptic action potential and the second postsynaptic action potential is varied from –100 ms up to 100 ms. Causal presynaptic-postsynaptic pairings correspond to positive T. Negative T denotes anti-causal postsynaptic–presynaptic pairings. The protocol of pairing is repeated 30 and 100 times at 5 Hz and at 0.5 Hz. In addition, in separate simulations a presynaptic action potential is paired with a single postsynaptic spike with varying temporal difference T, 30 and 100 times at 5 Hz. If the presynaptic action potential precedes the second postsynaptic spike by T = 10 ms, the depolarisation provided by the back-propagating spikes in the proximal SR region (Fig. 3a, black solid line) together with the presynaptically released glutamate opens NMDAr-gated channels, and the calcium concentration in the SR spine reaches a high level of 3.8 mM (Fig. 3b, solid line). 100 such causal pairings at 5 Hz results in CaMKII phosphorylation that remains active after 20 s of stimulation even when calcium returns to the resting levels for the following 400 s due to presynaptic and postsynaptic inactivity (Fig. 4a, solid line). PP2A is also activated but inhibited by CaMKII (Fig. 4b, solid line).

Fig. 3. Membrane potential and calcium concentration in a proximal SR spine during a single pre-post pairing. (a) Membrane potential in spine (black solid line: T = 10 ms; black dashed line: T = 10 ms) and in soma (gray line). The presynaptic action potential is induced at 20 ms when T = 10 ms, or at 40 ms when T = 10 ms. Postsynaptic action potentials are generated at 20 and 30 ms (2-spike burst). A second postsynaptic action potential is used as a reference point for T definition. (b) Calcium concentration in spine (solid line: T = 10 ms; dashed line: T = 10 ms).

AMPAr is potentiated and so the synaptic AMPA weight change is positive (Fig. 4c, solid line). If the temporal order of the pairings is reversed, the presynaptic action potential follows the second postsynaptic action potential. For T = 10 ms, the depolarisation provided by the back-propagating spikes in the SR region decreases by the time NMDAr is activated (Fig. 3a, dashed line), therefore the peak calcium concentration is low, approximately 1.2 mM (Fig. 3b, dashed line). This weak calcium signal is not sufficient to activate CaMKII (Fig. 4a, dashed line) after 100 anti-causal pairings at 5 Hz. Although CaMKII activity increases it does not stay phosphorylated after 20 s of stimulation when calcium returns to the resting levels. On the contrary, PP2A is dephosphorylated (active) and inhibits CaMKII (Fig. 4b, dashed line). The synaptic AMPA weight change is negative (Fig. 4c, dashed line) and AMPAr is depressed. LTP still occurs with only 30 causal pairings with T = 10 ms (Fig. 4d–f, solid lines). However, 30 anti-causal pairings fail to induce LTD and leave the synapse unmodified (Fig. 4d–f, dashed lines), as the calcium signal is not sufficient to dephosphorylate PP2A (Fig. 4f, dashed line). Synaptic weight changes and peak calcium concentrations in the proximal SR spine for temporal difference T values [100 ms, 100 ms] are presented in Fig. 5. High peak calcium concentrations above 2.5 mM are observed for causal pairings at 5 Hz, specifically for the positive T window from 0 ms up to 20 ms (Fig. 5a) and result in AMPAr potentiation (Fig. 5b, black line). Lower calcium concentrations below 2.5 mM and above 1 mM lead to AMPAr depression: LTD is obtained for anti-causal pairings within the T interval [50 ms,10 ms], and for causal pairings within the T interval [30 ms, 50 ms]. AMPAr stays at its basal level if the peak calcium concentration is low. The sombrero-shaped curve of synaptic modifications, however, is obtained only if the stimulation consists of 100 pairings at 5 Hz. 30 pairings at 5 Hz abolishes LTD and results in a potentiation-only plasticity rule (Fig. 5b, gray line). In contrast, decreasing the frequency of stimulation to 0.5 Hz leads to a depression-only learning rule for both 30 and 100 pairings (Fig. 5c). It is clear from Fig. 5 that the weight change curve (Fig. 5b and c) is not a simple function of peak calcium, but also depends on temporal factors such as the number of repetitions and frequency of the pairing protocol.

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Fig. 4. Activity of CaMKII (a and d), PP2A (b and e) and the resulting AMPAr weight changes (c and f) in a proximal SR spine after 20 s of stimulation with 100 pairings (a–c) and 6 s of stimulation with 30 pairings (d–f) at 5 Hz of 1 presynaptic and 2 postsynaptic action potentials, followed by 180 s of presynaptic and postsynaptic inactivity of a CA1 pyramidal neuron (c, solid line: T = 10 ms, AMPAr is potentiated; dashed line: T = 10 ms, AMPAr is depressed after 100 pairings of stimulation; f, solid line: T = 10 ms, AMPAr is potentiated; dashed line: T = 10 ms, AMPAr remains at a basal level after 30 pairings of stimulation).

Synaptic modifications are influenced by the pattern of postsynaptic activity as well. A single postsynaptic action potential, paired with a presynaptic action potential 100 times at 5 Hz, evokes depression, and 30 such pairings fail to induce synaptic changes (Fig. 6b, black and gray line, respectively) as the calcium influx is reduced (Fig. 6a). These simulated synaptic weight changes for causal and anticausal pairings of presynaptic and postsynaptic action potentials correspond to physiologically observed results (Wittenberg and Wang, 2006). The validated model of synaptic plasticity is used to estimate synaptic weight changes under various conditions and stimulation patterns. The same stimulation protocol consisting of 100 and 30 pairings of a presynaptic action potential with a doublet of postsynaptic action potentials at 5 Hz was applied not only for the proximal, but also for the medial and distal SR synapses. Medial and distal SR spines have the same volume as the proximal SR spine. Synaptic weight changes are presented in Fig. 7. Medial SR synapse, located 121 mm from the soma and having the same synaptic strength as the proximal SR synapse, undergoes LTP for causal pairings and LTD for anti-causal and causal pairings within the very narrow T intervals (Fig. 7a, black line). The decreased temporal width of the STDP curve is due to the fact that the back-propagating spike reaches the medial SR spine with a lower amplitude and results in reduced calcium influx both for anti-causal and causal pairings. Calcium levels at long, causal T are no longer sufficient to generate a causal LTD window. Distal SR spine 157 mm from the soma receives even weaker depolarisation by the back-propagating spike, therefore calcium levels are not sufficient to induce synaptic changes for anti-causal pairings, and only LTD is observed within a limited T interval [0 ms, 20 ms] for 100 pairings (Fig. 7c, black line). However, an increased AMPA synaptic weight reintroduces anticausal and causal LTD and LTP (Fig. 7d, black line) or widens the

observed LTP and LTD windows (Fig. 7b, black line) as intracellular calcium concentration rises to the levels necessary for LTD and LTP induction. The results indicate that synaptic modifications vary along the dendritic tree and depend on the synapse location and synaptic strength. In SR dendritic synapse location is encoded in the amplitude and time course of the back-propagating spike and the induced calcium levels. The synapse on the proximal SR spine is modified according to the wide sombrero-shaped STDP curve and undergoes both LTD and LTP. The synapse on the medial SR spine shows a narrow STDP temporal window with the reduced LTD and LTP parts. The synapse on the distal SR spine shows even narrower STDP window. However, as soon as the synapse becomes strong, it can undergo LTP and LTD within a wider T interval again. 3.1.2. Plasticity rule at excitatory synapse in SLM dendritic region Excitatory synapses in distal SLM dendrites of a CA1 pyramidal neuron are largely influenced by local dendritic regenerative action potentials but not much by somatic action potentials, as these fail to invade distal dendritic regions (Golding et al., 2002; Froemke et al., 2005; Letzkus et al., 2006). Changes at the synapse on a distal SLM spine thus are modelled by pairing a local dendritic spike as a source of postsynaptic depolarisation with the presynaptic action potential at varying temporal difference T values. The dendritic spike is induced by suprathreshold dendritic current injection. This spike consists of a sodium spike followed by a high-amplitude calcium spike (Fig. 8a) that provides depolarisation for NMDArgated channels even when anti-causal pairings are applied (Fig. 8b). Causal pairings at T = 10 ms induce high calcium influx and cause LTP (Fig. 8a and b, black solid lines; Fig. 9a, T = 10 ms). However, larger temporal difference T values in the interval [20 ms, 100 ms] lead to LTD as the postsynaptic sodium spike is not strong enough to effectively open NMDAr-gated channels and the

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Fig. 5. Synaptic modifications in a proximal SR synapse evoked by pairing a presynaptic action potential with a postsynaptic burst at 5 Hz and 0.5 Hz. (a) Peak calcium concentration in spine. (b) Synaptic weight change as a function of T, temporal difference between the presynaptic and a second postsynaptic action potential, after 100 pairings (black line) and 30 pairings (gray line) at 5 Hz, followed by 400 s of presynaptic and postsynaptic inactivity. (c) Synaptic weight change after 100 pairings (black line) and 30 pairings (gray line) at 0.5 Hz, followed by 400 s of presynaptic and postsynaptic inactivity.

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subsequent calcium spike arrives too late, but moderate depolarization at the postsynaptic site is provided by a dendritic spike of the preceding pre- and postsynaptic spike pair. Consequently, the synapse is weakened for 100 pairings and remains unmodified for

Fig. 7. Synaptic weight change in the medial and distal SR synapses as a function of T, temporal difference between the presynaptic and a second postsynaptic action potential, after 100 pairings (black line) and 30 pairings (gray line) at 5 Hz, followed by 400 s of presynaptic and postsynaptic inactivity. (a) Medial SR spine is located 121 mm from the soma, AMPA synaptic weight is 2.2 nS; (b) Medial SR spine is located 121 mm from the soma, AMPA synaptic weight is 13 nS; (c) Distal SR spine is located 157 mm from the soma, AMPA synaptic weight is 2.2 nS; (d) Distal SR spine is located 157 mm from the soma, AMPA synaptic weight is 13 nS.

Fig. 6. Synaptic modifications in a proximal SR synapse evoked by pairing a presynaptic action potential with a single postsynaptic action potential (AP) at 5 Hz. (a) Peak calcium concentration in spine. (b) Synaptic weight change as a function of T, temporal difference between the presynaptic and a postsynaptic action potential, after 100 pairings (black line) and 30 pairings (gray line), followed by 400 s of presynaptic and postsynaptic inactivity.

Fig. 8. Membrane potential and calcium concentration in a SLM spine. (a) Membrane potential in spine (black solid line: T = 10 ms; black dashed line: T = 10 ms) and in soma (gray line). The presynaptic action potential is induced at 10 ms (black solid line: T = 10 ms) or at 30 ms (black dashed line: T = 10 ms), the postsynaptic dendritic spike is induced at 20 ms by suprathreshold dendritic current injection. (b) Calcium concentration in spine (solid line: T = 10 ms; dashed line: T = 10 ms).

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Fig. 9. Synaptic modifications in a SLM spine evoked by pairing a presynaptic action potential with a dendritic spike at 5 Hz. (a) Peak calcium concentration in spine. (b) Synaptic weight change as a function of T, temporal difference between the presynaptic action potential and the onset of dendritic spike, after 100 pairings (black line) and 30 pairings (gray line), followed by 400 s of presynaptic and postsynaptic inactivity. The postsynaptic dendritic spike is induced by suprathreshold dendritic current injection.

30 pairings at 5 Hz in the T interval [20 ms, 100 ms] (Fig. 9b, black and gray lines, respectively). For anti-causal pairings and T = 10 ms, NMDAr-gated channels are strongly depolarized by the dendritic calcium spike, calcium levels rises to 7.8 mM (Fig. 8a and b, black dashed lines) and 100 and 30 such pairings at 5 Hz lead to LTP. Synaptic weight changes and peak calcium concentration in the SLM spine for temporal difference T values are shown Fig. 9. Peak calcium concentration is high for negative T and leads to LTP in a wide T interval [30 ms, 10 ms] and to LTD in the T intervals [100 ms, 40 ms] and [20 ms, 100 ms] for 100 pairings at 5 Hz. Decreasing the number of pairings to 30 abolishes LTD and results in a LTP-only synaptic plasticity rule. These results show that synapses in distal SLM dendritic regions of the CA1 pyramidal neuron are potentiated if activated shortly before or during the induction of a long-lasting dendritic spike, and depressed if triggered once the dendritic membrane potential has returned to the moderately depolarized level for anti-causal and causal pairings. The temporal order of the pre- and local postsynaptic events is neglected. Timing requirements for LTP are not as strict as for the plasticity rule of excitatory synapses in the SR proximal dendritic region which shows a sharp transition from LTD to LTP at 0 ms and has a narrow LTP window (Fig. 5b). 3.2. Influence of inhibitory interneurons Pyramidal neurons in hippocampal CA1 regions are inhibited by spatially targeted perisomatic and dendritic inhibitory inputs (Klausberger et al., 2003; Cutsuridis et al., 2010). Specifically, basket interneurons provide perisomatic inhibition, while dendritic inhibition is induced by bistratified and OLM interneurons. 3.2.1. Plasticity rule at excitatory synapse in SR dendritic region in the presence of perisomatic inhibition provided by a basket cell To reveal the influence of the perisomatic inhibitory interneurons on the plasticity in SR synapses, the pyramidal neuron is inhibited by a basket cell. This interneuron is activated by somatic spiking of the pyramidal cell (PC) (Cutsuridis et al., 2010), inhibits the soma of the pyramidal neuron and prevents the generation of the second action potential in the two spike burst (Fig. 10a, gray line). The excitatory synapse of the CA1 pyramidal neuron–basket cell connection is sufficiently strong that the basket cell generates

Fig. 10. Membrane potential and calcium concentration in a proximal SR spine under the inhibitory effect of a basket cell BC. (a) Membrane potential in spine (black solid line: T = 10 ms; black dashed line: T = 10 ms) and in soma (gray line) of the pyramidal cell PC. The presynaptic action potential is induced at 20 ms (black solid line: T = 10 ms) or at 40 ms (black dashed line: T = 10 ms). The second postsynaptic spike at 30 ms is a dummy, not generated due to inhibition, and used as a reference point for T definition. (b) Membrane potential in soma of a basket cell BC. (c) Calcium concentration in a SR spine (solid line: T = 10 ms; dashed line: T = 10 ms). Soma of the CA1 pyramidal neuron is inhibited by a basket cell that is activated by the first somatic action potential of the pyramidal neuron PC.

three action potentials in response to the PC spike (Fig. 10b). As a result of the reduced PC somatic spiking activity, the membrane potential at the SR synapse location and calcium concentration in the SR spine are considerably reduced for pre-post and post-pre pairing protocols (Fig. 10a and c, T = 10 ms and T = 10 ms, black solid line and black dashed line, respectively). For T = 10 ms, the calcium concentration in the SR spine rises to 1.5 mM (Fig. 10b, solid line) and only to 0.6 mM for T = 10 ms (Fig. 10c, dashed line). Calcium levels are lower than the calcium concentration observed for a single post-synaptic action potential of pyramidal cell (Fig. 6a, solid line, T = 10 ms, T = 10 ms) due to effective perisomatic inhibition. Peak calcium concentration in the SR spine and synaptic modifications after 30 and 100 pairings at 5 Hz for temporal difference T values are shown in Fig. 11. Peak calcium concentration is reduced at both negative and positive T values (Fig. 11a), and compares to the peak calcium levels obtained using a single postsynaptic action potential (Fig. 6a). The synapse is not modified if the presynaptic action potential arrives once the inhibition is already triggered for anti-causal pairings. The basket interneuron not only blocks a second post-synaptic action potential, but effectively hyperpolarizes soma for the tens of milliseconds, prevents depolarization of a proximal spine by a single backpropagating spike and thus reduces calcium influx through an NMDAr-gated channel. The synapse is depressed if the presynaptic activation shortly precedes the somatic inhibition for 100 pairings (Fig. 11b, black line). However, 30 pairings does not induce synaptic modifications (Fig. 11b, gray line). Synaptic changes under the influence of a basket interneuron resemble the results obtained applying a single post-synaptic action potential (Fig. 6b), as only LTD is induced.

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narrow temporal window for long stimulation. Synapses remain unmodified for short stimulation protocols.

Fig. 11. Synaptic modifications in a proximal SR synapse evoked by pairing a presynaptic action potential with a doublet of postsynaptic action potentials at 5 Hz under the inhibitory effect of a basket cell. (a) Peak calcium concentration in spine. (b) Synaptic weight change as a function of T, temporal difference between the presynaptic action potential and a dummy second postsynaptic spike (not generated due to inhibition) after 100 pairings (black line) and 30 pairings (gray line) at 5 Hz, followed by 400 s of presynaptic and postsynaptic inactivity. Soma of the CA1 pyramidal neuron is inhibited by a basket cell that is activated by the first somatic action potential of the pyramidal neuron.

3.2.2. Plasticity rule at an excitatory synapse in the SR dendritic region in the presence of dendritic inhibition provided by a bistratified interneuron Synapses at SR dendrites of CA1 pyramidal cells are affected not only by perisomatic basket inhibition that may block somatic spiking, but also by bistratified interneurons that inhibit SR dendrites. Bistratified cells receive CA3 inputs that also stimulate excitatory synapses on SR dendrites (Cutsuridis et al., 2010). In this study, the bistratified neuron is triggered by the same presynaptic CA3 input that activates the SR synapse. Strong dendritic inhibition impairs back-propagation of somatic action potentials and consequently the membrane potential at the proximal SR synapse is reduced (Fig. 12a, compare with Fig. 3a). The calcium concentration in the spine now hardly reaches 1.8 mM for T = 10 ms (Fig. 12c, black solid line) resulting in LTD (Fig. 13b, black line, T = 10). Causal pairing with increasing T leads to LTD within the T interval [0 ms; 30 ms] (Fig. 13b, black line, positive T). However, the membrane potential at the synapse and the calcium concentration are only slightly affected by inhibition for anticausal pairings (Fig. 12a and c, black dashed lines, T = 10 ms) as the bistratified cell is activated too late to effectively inhibit the back-propagating action potentials (Fig. 12b, dashed line, T = 10 ms). Thus the acausal LTD window with 100 pairings is retained

Perisomatic inhibition, mediated by basket interneurons, refines excitatory synaptic modifications by terminating somatic burst firing, limiting calcium influx into a proximal spines, and leads to depression of the synapse if it is causally activated within a

Fig. 12. Membrane potential and calcium concentration in a proximal SR spine under the inhibitory effect of a bistratified cell BSC. (a) Membrane potential in spine (black solid line: T = 10 ms; black dashed line: T = 10 ms) and in soma (gray solid line: T = 10 ms; gray dashed line: T = 10 ms). The presynaptic action potential is induced at 20 ms (solid line: T = 10 ms) or at 40 ms (dashed line: T = 10 ms). (b) Membrane potential in soma of a bistratified interneuron BSC. (c) Calcium concentration in SR spine (solid line: T = 10 ms; dashed line: T = 10 ms). SR dendritic region of the CA1 pyramidal neuron is inhibited by a bistratified interneuron that is activated by the same CA3 input as the SR synapse. Bistratified inhibition is strong (gGABAA ¼ 10 nS, gGABAB =1 nS).

Fig. 13. Synaptic modifications in a proximal SR synapse evoked by pairing a presynaptic action potential with a doublet of postsynaptic action potentials at 5 Hz under the inhibitory effect of a bistratified cell. (a) Peak calcium concentration in spine: strong dendritic inhibition (solid line: gGABAA ¼ 10 nS, gGABAB =1 nS) and weak dendritic inhibition (dashed line: gGABAA ¼ 5 nS, gGABAB =0.5 nS) (b) Synaptic weight change as a function of T, temporal difference between the presynaptic action potential and a second postsynaptic spike, after 100 pairings (black line) and 30 pairings (gray line), followed by 400 s of presynaptic and postsynaptic inactivity; dendritic inhibition is strong. (c) Synaptic weight change as a function of T after 100 pairings (black line) and 30 pairings (gray line), followed by 400 s of presynaptic and postsynaptic inactivity; dendritic inhibition is weak. SR dendritic membrane potential of the CA1 pyramidal neuron is inhibited by a bistratified interneuron that is activated by the same CA3 input as the SR synapse.

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but is narrower than with no inhibition (Fig. 13b, black solid line, negative T; compare with Fig. 5b). In summary, strong dendritic inhibition results in a depressiononly plasticity rule for 100 pairings at 5 Hz (Fig. 13b, black line) and prevents synaptic changes for 30 pairings at 5 Hz (Figure 13b, gray line). If the inhibition is weakened, then the shape of the STDP curve remains symmetrical for 100 pairings, but the causal LTP window is narrowed and the causal LTD side window is consequently left-shifted towards smaller values of T, compared with no inhibition (Fig. 13c, black line; compare with Fig. 5b, black line). Short stimulation of 30 pairings abolishes LTD and leads to LTP in a temporal window [0; 20 ms] (Fig. 13c, gray line). 3.2.3. Plasticity rule at an excitatory synapse in the SLM region in the presence of SLM dendritic inhibition provided by an OLM interneuron The distal SLM dendrites are affected by inhibition that is triggered by OLM cells (Cutsuridis et al., 2010). In the model, OLM interneurons are activated by the CA1 pyramidal neuron somatic action potential induced by the SLM dendritic spike approximately 25 ms after its onset (Fig. 14a and b). Thus OLM inhibition reduces depolarization at the synapse location and calcium levels in the SLM spine with a 25 ms delay in respect to the postsynaptic activity (dendritic spike). Due to the delayed activation, the OLM cell is not effectively influencing membrane potential and calcium levels at the SLM spine for pairings at T = 10 ms and T = 10 ms (Fig. 14a and c, black solid and dashed lines, respectively; see Fig. 8). Calcium concentration is slightly affected by the OLM inhibition within the T interval [20 ms; 20 ms] and it is decreased for the remaining T values (Fig. 15a; see Fig. 9a). For the negative T values calcium

Fig. 15. Synaptic modifications in a SLM spine evoked by pairing a presynaptic action potential with a dendritic spike at 5 Hz under the inhibitory effect of an OLM cell. (a) Peak calcium concentration in spine. (b) Synaptic weight change as a function of T, temporal difference between the presynaptic action potential and the onset of dendritic spike, after 100 pairings (black line) and 30 pairings (gray line), followed by 400 s of presynaptic and postsynaptic inactivity. The postsynaptic dendritic spike is induced by suprathreshold dendritic current injection. SLM dendritic region of a pyramidal neuron is inhibited by an OLM cell triggered by the somatic action potentials of the pyramidal cell with approximately a 25 ms delay after the onset of the dendritic spike.

levels are reduced due to the OLM inhibition triggered by a dendritic spike of the preceding pre- and postsynaptic spike pair. Therefore, synaptic depression is abolished for anti-causal pairings and synaptic potentiation is observed within the T interval [20 ms; 10 ms] for long (100 pairings) stimulation (Fig. 15b, black line). Short (30 pairings) stimulation leads to synaptic strengthening within the T interval [20 ms; 10 ms] and no synaptic modification for the remaining T values (Fig. 15b, gray line), which is very similar to when no inhibition is present (Fig. 9b, gray line). Thus, OLM inhibition removes LTD at acausal and long causal time intervals, without causing much effect on LTP at shorter acausal or any causal time intervals, when it is activated in response to a SLM dendritic spike. 4. Discussion 4.1. Summary This modeling study demonstrates explicitly the effects of variation in spine calcium due to synaptic location and the presence of spatially and temporally specific inhibition on synaptic plasticity. Both LTP and LTD windows in spike time dependent plasticity curves are strongly affected by these factors. The results confirm and extend experimental data and other modeling work on this issue. 4.2. Temporal requirements of plasticity

Fig. 14. Membrane potential and calcium concentration in a SLM spine under the inhibitory effect of an OLM cell. (a) Membrane potential in spine (black solid line: T = 10 ms; black dashed line: T = 10 ms) and in soma (gray line). The presynaptic action potential is induced at 10 ms (black solid line: T = 10 ms) or at 30 ms (black dashed line: T = 10 ms), the postsynaptic dendritic spike is induced at 20 ms by suprathreshold dendritic current injection. (b) Membrane potential in soma of an OLM interneuron spine (solid line: T = 10 ms; dashed line: T = 10 ms). (c) Calcium concentration in SLM spine (solid line: T = 10 ms; dashed line: T = 10 ms). SLM dendritic region of the CA1 pyramidal neuron is inhibited by an OLM cell triggered by the somatic action potentials of the CA1 pyramidal cell with approximately a 25 ms delay after the onset of the dendritic spike.

An important aspect of synaptic plasticity that is not considered in most models is the distinct temporal requirements for inducing LTP and LTD, such as revealed by the experimental data of Wittenberg and Wang (2006). Not only is LTD induced by lower calcium levels than LTP, it also requires many more repetitions of this lower calcium. This implies the signaling pathways underpinning LTD have slower dynamics than those for LTP. These distinct dynamics can be captured in models that employ competitive LTP and LTD pathways in which large calcium fluxes will rapidly activate the LTP pathway, which then “wins” over LTD. Lower

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calcium fluxes allow the slower activation of the LTD pathway which is no longer inhibited by the only weakly activated LTP pathway. Recent models successfully capture this behavior in simple ways (Bush and Jin, 2012; Graupner and Brunel, 2012). One model (Bush and Jin, 2012) is based on kinase and phosphatase dynamics expressed in an abstract way. Kinase and phosphatase activities are controlled by spine calcium concentration. Raised kinase activity increases the probability that a synapse will transition from a low to a high strength, and inhibits phosphatase activity. Conversely, phosphatase activity increases the probability of a high to low synaptic transition. Graupner and Brunel (2012) reduce this process to the dynamics of a single variable that is driven by spine calcium and determines binary synaptic strength (either low or high). A weight transition depends on the starting state of the synapse, so that high weight synapses can only undergo LTD, and vice versa. Here we demonstrate this phenomenon with a more detailed tristate model in which the competition of kinase and phosphatase is described by complex equations that reflect the dynamics of the molecular networks underlying synaptic plasticity (Graupner and Brunel, 2007; Pi and Lisman, 2008). The model includes the molecular network of CaMKII activation (Graupner and Brunel, 2007), responsible for LTP induction, and in addition incorporates an LTD mechanism based on PP2A activation (Pi and Lisman, 2008). The tristability of the Pi and Lisman (2008) model is retained by including their terms for constant, basal kinase and phosphatase activity levels. The tristability is not crucial to the results, but allows the straightforward examination of plasticity induction protocols at a synapse, since the synapse can undergo either LTP, LTD, or no change when starting from a single basal state. Binary models (Bush and Jin, 2012; Graupner and Brunel, 2012) require examining synapses starting in either low or high states. The limited experimental evidence on plasticity at individual synaptic connections seems to indicate that synapses indeed are binary (O’Connor et al., 2005), but further work is needed for this to be conclusive. The obtained model realizes the temporal characteristics of the biochemical reactions that describe synaptic modifications. Therefore, the model is sensitive to the amplitude and time course of intracellular calcium transients. It is able to simulate synaptic changes for experimentally used stimulation protocols, where the timing of pre- and post-synaptic spiking, pattern of postsynaptic activity, frequency and duration of stimulation affects the calcium transients in individual dendritic spines where synapses are formed. We have tuned the model to reproduce the data of Wittenberg and Wang (2006). The match of the Bush and Jin (2012) model to this data partially arises from the fact that maximal calcium concentration depends on spike pairing frequency, even for low frequencies of 0.5–5 Hz. The duration of stimulation scales the amplitude of synaptic modifications, but does not change the sign. In our model, maximal spine calcium concentration remains the same for frequencies of 0.5–5 Hz, regardless of the number of spike pairings. Synaptic modification dependence on frequency and duration of spike pairings emerges due to the modeled molecular network sensitivity to the overall calcium concentration time course, and the obtained synaptic changes can be opposite in sign for short and long stimulations (e.g., Fig. 7a, T = 0 ms). Bush and Jin (2012) modeled spines with NMDAr-channel mediated current as a single source of calcium entry and used an artificially produced backpropagating somatic action potential. We use a detailed compartmental model of a CA1 pyramidal neuron, with back-propagating spikes produced by the active membrane properties of the cell, and so have realistic shapes. In addition to NMDAr, the spines have voltage sensitive calcium channels, which contribute to the calcium magnitude and time course.

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4.3. Influence of synaptic location We use the detailed synaptic plasticity model within a detailed cell model to examine how STDP properties are influenced by a synapse's dendritic location and the activity of inhibitory cells impinging on a CA1 pyramidal neuron. STDP curves show a significant dependence on dendritic synapse location. Synaptic modifications at SR synapses strongly depend on the exact timing of the presynaptic action potential and postsynaptic back-propagating spikes. Proximal synapses are potentiated by small and large numbers of causal pre-post pairings (Fig. 5b). They are depressed by a large number of anti-causal post-pre pairings, and during a short causal time window when the back-propagating spikes correspond with the tail of the synaptic NMDA current (Fig. 5b). Symmetric and asymmetric STDP curves have been produced with the original Pi and Lisman (2008) model, with calcium entry through voltage dependent calcium channels implicated in the removal of the causal LTD window (Carlson and Giordiano, 2011). Attenuation of the back-propagating spike amplitude with distance lowers the peak calcium amplitudes obtained in medial and distal spines. If these spines have the same initial strength as a proximal spine, then this lowering of peak calcium leads to the narrow LTP and LTD windows at the medial SR synapse (Fig. 7a) and results in the causal LTP window seen proximally changing to LTD, and the proximal LTD side windows disappear at the distal SR synapse (Fig. 7c). Increasing the AMPA conductance widens LTP window with LTD side windows at the medial synapse (Fig. 7b), and reintroduces LTP window at the distal synapse (Fig. 7d). Similar changes with distance have been seen experimentally, in particular the disappearance of the causal LTD window, which may be most sensitive to a reduction in spine calcium (Tsukada et al., 2005). At the distal synapses, a wider LTP window can be obtained if the LTP threshold of the molecular network underlying synaptic plasticity is lowered or calcium levels in the spine are increased through reduced spine volume and other spine shape regulating mechanisms (results not shown). Triggering of a dendritic spike in this region should also result in LTP, as for SLM synapses. The influence of dendritic location also has been analyzed in the work of Graupner and Brunel (2012), and it was shown that LTP turns into LTD at distal synapses, but strong depolarization can rescue LTP. Back-propagating somatic spikes have no influence in the SLM region, hence we explore the STDP curve when the postsynaptic activity is a locally initiated slow dendritic spike, mediated by sodium and calcium. The contribution of distal dendritic spikes to LTP at these synapses is established (Golding et al., 2002) but the precise form of the STDP curve has not yet been measured experimentally. Using the start of the dendritic spike as the reference time for postsynaptic activity for calculating pre-post time differences means the LTP window is largely anti-causal because the dendritic spike only reaches its peak tens of milliseconds after its onset, thus favoring presynaptic spikes arriving after its onset. There is a small causal LTP window, so synapses receiving input activity that results in a dendritic spike will undergo LTP, as well as inducing potentiation at other SLM synapses active up to 40 ms later. This time window encompasses at least the next cycle in gamma frequency activity, which is prominent in exploring animals such as rats (Leung et al., 1982). This may allow a sufficiently broad sampling of grid cell activity for the formation of “place” receptive fields from the SLM inputs to the CA1 pyramidal cell (Savelli and Knierim, 2010). Sufficient pairings (100) of presynaptic spikes with the tail of the dendritic spike produces a wide acausal and causal LTD window, so the temporal sampling of entorhinal cortex activity is strictly limited by the duration of the dendritic spike.

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4.4. Influence of inhibition In addition to site-specificity, perisomatic and dendritic inhibition shape the plasticity rules at SR and SLM synapses. Whereas other models consider the effects of essentially steadystate inhibition present throughout excitation (Bar-Ilan et al., 2013), we consider the implications of particular timings of inhibition associated with excitatory inputs. Inhibition reduces synaptic calcium levels, particularly at causal times with respect to presynaptic inputs, if the inhibition is driven directly by these same inputs (feedforward inhibition) or in a time window following postsynaptic firing if driven by this activity (feedback inhibition). Perhaps counter-intuitively, inhibition can promote synaptic LTP, because LTD occurs for moderate calcium levels and is thus more vulnerable to any reduction in peak calcium. In our simulations, at SR synapses, feedback basket interneuron-induced perisomatic inhibition reduces somatic burst firing in the pyramidal cell to a single spike. The consequent reduction in depolarisation at the SR synapse blocks LTP and leaves a narrow, causal LTD window (Fig. 11b). Thus this form of feedback inhibition allows spiking cell output in response to excitatory input while promoting depression of those active inputs. Feedforward bistratified interneuron-mediated inhibition is strongest at early causal times and consequently narrows the causal LTP window, and shifts LTD to earlier causal and acausal times (Fig. 13c). Thus it effectively refines the temporal constraints on synaptic plasticity. Strengthening this inhibition can remove the causal LTP window and leave only LTD (Fig. 13b). At SLM synapses, feedback inhibition from OLM neurons results basically in a potentiation-only plasticity rule, even for many pairings (Fig. 15b). This feedback inhibition arrives towards the end of the dendritic spike and is thus strongest during the time window in which acausal LTD occurs in the absence of inhibition (Fig. 9b). These results are consistent in principle with experimental data on the effects of inhibition on plasticity. Tsukada et al. (2005) show that inhibition may be responsible for the causal LTD window at proximal SR synapses. In their experiments, a causal time point that gave LTD in the presence of inhibition reverted to no change in synaptic strength when inhibition was blocked. In other words, inhibition at that time point reduced synaptic calcium from a level that was high but not quite sufficient to induce LTP to a moderate level that gave LTD. Cutsuridis (2011, 2012),) has demonstrated this effect using a phenomenological plasticity model with long-lasting inhibition timed with respect to pre-post spike pairs. Our model is tuned without inhibition to reproduce the symmetric, bidirectional STDP curve observed by Wittenberg and Wang (2006) even when inhibition is blocked. Now adding dendritic inhibition (in the form of feedforward inhibition from bistratified cells) both changes causal LTP into LTD and causal LTD into no change (Fig. 13b) due to the reduced spine calcium levels at causal times. So this inhibition promotes LTD at early causal times (as in Tsukada et al., 2005), while reducing it at longer causal times. These modeling results demonstrate that, depending on its timing and strength, inhibition can (1) refine the temporal requirements for plasticity by narrowing the LTP and LTD time windows, (2) protect synapses and promote LTP by removing LTD side windows, or (3) promote LTD by changing LTP windows into LTD windows. This depends on the exact calcium levels that inhibition is acting on, which may be moderate or high, respectively. Here we have considered spatially focused inhibition in register with the excitatory inputs. In a modeling study of plasticity in a neocortical pyramidal neuron, Bar-Ilan et al. (2013) also consider the spread of focused inhibition along dendrites, and demonstrate that individual synapses within a population of active synapses may be protected (no change), potentiated or depressed, depending on their spatial relationship to the inhibitory input. In a

purely phenomenological model it was shown that feedback inhibition narrows a LTD window in a cortical microcircuit (Porr et al., 2011). All of these modeling results support the notion that not only timing of pre- and postsynaptic activity, but also location of the synapse and spatially-targeted inhibition are important determinants of STDP (Froemke et al., 2005; Letzkus et al., 2006).

4.5. Models of synaptic plasticity Computational models of synaptic plasticity can be divided into two broad classes: phenomenological and biophysical. Phenomenological models generally link neuronal spiking or firing rate activity directly to synaptic changes via the dynamics of abstract quantities that are not readily related to particular cellular mechanisms (Gerstner et al. 1996; Kempter et al., 1999; Kistler and van Hemmen, 2000; Song et al., 2000; Saudargiene et al., 2004; Pfister and Gerstner, 2006). Biophysical models attempt to describe the calcium dynamics and protein signaling cascades underlying synaptic modifications. They tend to rely on abstractions as well as kinetic descriptions of intracellular signalling, leading to models of different complexity. For example, abstract biophysical models are based on the calcium control hypothesis, stating that a large calcium transient induces LTP and a moderate elevation in calcium leads to LTD (Shouval et al., 2002; Shouval and Kalantzis, 2005; Rackham et al., 2010). Slighly more detailed models include simplified descriptions of intracellular processes, typically kinase and phosphotase activities, and may be termed reduced biophysical models (e.g., Abarbanel et al., 2002; Pi and Lisman, 2008; Carlson and Giordiano, 2011; Bush and Jin, 2012; Graupner and Brunel, 2012). Detailed models of such signaling network activation can be very complex (Bhalla and Iyengar, 1999; Zhabotinsky, 2000; Graupner and Brunel 2007; Urakubo et al., 2008; see Graupner and Brunel, 2010; Kotaleski and Blackwell, 2010; Manninen et al., 2010; Shouval et al., 2010 for review). The distinction between biophysical and phenomenological models is not absolute. All current biophysical models include mechanisms described at different levels of abstraction, including such approximations as Michaelis–Menten and Hill functions. Though these models make an explicit attempt to identify quantities and pathways in the model with known molecules and biochemical reactions, the dynamical equations employed are not always derivable from the basic principles of chemical interactions and are generally chosen so that the model can match particular sets of experimental data on STDP. This is the approach we have taken here, and the resultant model is able to reproduce complex dynamics of plasticity revealed by experiments (Wittenberg and Wang, 2006).

5. Conclusion Studies such as this, conducted with computational models of synaptic plasticity and properties of STDP in CA1 pyramidal neurons, contribute to the understanding of molecular, cellular and network mechanisms underlying long-term memory formation in neural structures. Conflict of interest The authors declare that they have no conflicts of interest in the research.

_ B.P. Graham / BioSystems 130 (2015) 37–50 A. Saudargiene,

49

Authors’ contributions AS and BG designed the study, AS performed simulations. Both authors analyzed the results, wrote and approved the final manuscript. Acknowledgments We thank Marja-Leena Linne for helpful comments and fruitful discussions. This research was funded by a grant (No MIP-93/2010) from the Research Council of Lithuania.

d S3 ¼ 2k6 g 2 S1  2k6 g 2 S3  2k7 g S3  2kD S3 dt þ kD ðS5 þ S6 þS7 þ3S8 Þ;

d S4 ¼ k6 g 2 S1  2k6 g 2 S4  2k7 g S4  2kD S4 þ kD ðS6 þ S7 Þ; dt

(A. 11)

(A. 12)

Appendix A.

d S5 ¼ k6 g 2 S2 þ k7 g 2 ðS2 þ S3 Þ  2k6 g 2 S5  k7 g S5  3kD S5 dt þ kD ð2S9 þ S10 Þ;

(A. 13)

STDP model is decribed by the following set of equations. Parameter definitions and values are presented in Table A1. Concentration of calcium–calmodulin complex (Graupner and Brunel, 2007):

d S6 ¼ k6 g 2 ðS2 þ S3 Þ þ 2k7 g S4  k6 g 2 S6  2k7 g S6 dt  3kD S6 þ kD ðS9 þ S10 þ 2S11 Þ;

(A. 14)

d S7 ¼ k6 g 2 ðS2 þ S4 Þ þ k7 g S3  k6 g 2 S7  2k7 g S7  3kD S7 dt þ kD ðS9 þ S10 þ 2S11 Þ;

(A. 15)

CaM ¼

CaMTot 4 1 þ ½CaK2þ þ 

K3 K4 ½Ca2þ 2

K3 K4 þ K 2 K2þ3 K34 þ K 1 K 22þ 4 ½Ca



½Ca

:

(A. 1)



PKA activity (Graupner and Brunel, 2007):

nPKA ¼ koPKA þ

1þ

kPKA  nPKA : K PKA CaM

(A. 2) d S8 ¼ k6 g 2 S3  3k7 g S8  3kD S8 þ kD S10 ; dt

Calcineurin activity (Graupner and Brunel, 2007):

nCaN ¼ koCaN þ

1þ

kCaN  nCaN : K CaN CaM

(A. 3)

Concentration of active PP1 (Graupner and Brunel, 2007): d PP1 ¼ kPP1 I1P  PP1 þ kPP1 ðPP1Tot  PP1Þ: dt

(A. 4)

Concentration of active I1P (Graupner and Brunel, 2007): d I1P ¼ kPP1 I1P  PP1 þ kPP1 ðPP1Tot  PP1Þ þ nPKA I1PTot dt  nCaN I1P:

(A. 5)

Probability that CaMKII subunit binds with calcium-calmodulin complex (Graupner and Brunel, 2007):

g¼

CaM : K 5 þ CaM

(A. 6)

Rate of CaMKII subunit dephosphorylation (a modified equation from Graupner and Brunel (2007)): kD ¼

kPP1 PP1 þ kPP2A PP2A : K M þ CaMKII

(A. 7)

Concentrations of CaMKII with different numbers of phosphorylated subunits (Graupner and Brunel, 2007): d S0 ¼ 6k6 g 2 S0 þ kD S1 ; dt

(A. 8)

(A. 16)

d S9 ¼ k6 g 2 S5 þ k7 g ðS5 þ S6 þ S7 Þ  2k6 g 2 S9  2k7 g S9 dt  4kD S9 þ 2kD S12 ;

(A. 17)

d S10 ¼ k6 g 2 ðS5 þ S6 Þ þ k7 g ðS7 þ 3S8 Þ  2k7 g 2 S10 dt  4kD S10 þ 2kD S12 ;

(A. 18)

d S11 ¼ k6 g 2 S7 þ k7 g S6  2k7 g S11  4kD S11 þ 2kD S12 ; dt

d S12 ¼ k6 g 2 S9 þ k7 g ðS9 þ 2S10 þ 2S11 Þ  k7 g S12  5kD S12 dt þ 6kD S13 ;

d S13 ¼ k7 g S12  6kD S13 : dt

(A. 19)

(A. 20)

(A. 21)

Concentration of phosphorylated CaMKII subunits (Graupner and Brunel, 2007): CaMKII ¼ S1 þ 2ðS2 þ S3 þ S4 Þ þ 5ðS5 þ S6 þ S7 þ S8 Þ þ 4ðS9 þ S10 þ S11 Þ þ 5S12 þ 6S13 :

(A. 22)

Concentration of dephosphorylated PP2A (a modified equation from Pi and Lisman (2008)): d S1 ¼ 6k6 g 2 S0  4k6 g 2 S1  k7 g S1  kD S1 dt þ 2kD ðS2 þ S3 þ S4 Þ;

d S2 ¼ k6 g 2 S1 þ k7 g S1  3k6 g 2 S2  k7 g S2  2kD S2 dt þ kD ð2S5 þ S6 þ S7 Þ;

(A. 9)

(A. 10)

d PP2ATot  PP2A PP2A ¼ k11 PP2A dt K m11 þ PP2ATot  PP2A PP2A  k12 ðCaMKII þ CaMKII0 Þ K m12 þ PP2A ½Ca2þ 3 ðPP2ATot  PP2AÞ: þ k13 PP2A0 þ k14 3 K m þ ½Ca2þ 3

(A. 23)

50

_ B.P. Graham / BioSystems 130 (2015) 37–50 A. Saudargiene,

Rate of AMPAr phosphorylation (Pi and Lisman, 2008): kAMPA ¼ c1 CaMKII þ c2 :

(A. 24)

Rate of AMPAr dephosphorylation (Pi and Lisman, 2008): k AMPA ¼ c3 PP2A þ c4 :

(A. 25)

Concentration of active AMPAr (Pi and Lisman, 2008): d AMPA ¼ kAMPA ðAMPATot  AMPAÞ  kAMPA AMPA: dt

(A. 26)

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