Control of Ca2+ Influx and Calmodulin Activation by SK-Channels in Dendritic Spines

The key trigger for Hebbian synaptic plasticity is influx of Ca2+ into postsynaptic dendritic spines. The magnitude of [Ca2+] increase caused by NMDA-receptor (NMDAR) and voltage-gated Ca2+ -channel (VGCC) activation is thought to determine both the amplitude and direction of synaptic plasticity by differential activation of Ca2+ -sensitive enzymes such as calmodulin. Ca2+ influx is negatively regulated by Ca2+ -activated K+ channels (SK-channels) which are in turn inhibited by neuromodulators such as acetylcholine. However, the precise mechanisms by which SK-channels control the induction of synaptic plasticity remain unclear. Using a 3-dimensional model of Ca2+ and calmodulin dynamics within an idealised, but biophysically-plausible, dendritic spine, we show that SK-channels regulate calmodulin activation specifically during neuron-firing patterns associated with induction of spike timing-dependent plasticity. SK-channel activation and the subsequent reduction in Ca2+ influx through NMDARs and L-type VGCCs results in an order of magnitude decrease in calmodulin (CaM) activation, providing a mechanism for the effective gating of synaptic plasticity induction. This provides a common mechanism for the regulation of synaptic plasticity by neuromodulators.

In our model, NMDAR-and VGCC-dependent spine-calcium influx varies in response to pre-and postsynaptic spiking activity, the spike-timings of which act as the model input. Spine membrane potential, V m , is the sum of the membrane depolarization due to synaptically-evoked currents, V i , and back-propagating action potentials, V bAP .

Back-propagating action potentials
Depolarization due to bAPs is modeled with a double exponential, as in [1,2] V bAP (t) = where I bs and I b f set the relative contributions of the fast and slow components and t post is the array of all postsynaptic spike times. The total contribution to membrane depolarization is a summation over all t post , so bAPs in close temporal proximity have an additive effect.

Ion currents
The model includes the following currents: I A and I N mediated by AMPARs and NMDARs, I CaT and I CaL mediated by T-type and L-type VGCCs, a calcium-activated K + current, I SK , mediated by SK-channels, and a leak current, I L . VGCC currents are small relative to the other currents, so we assume negligible VGCC contribution to spine depolarization. VGCCs therefore act simply as voltage-dependent Ca 2+ sources. Using Hodgkin-Huxley formalism, the spine membrane potential due to ion channel currents, V i , evolves according to where C m is the membrane capacitance and c is [Ca 2+ ] local to the SK-channel. The AMPAR, NMDAR and leak ion currents are modeled as where r A and r N are the fraction of glutamate-bound receptors in the relevant synaptically-activated ion-channel cluster (calculated as in [3]), and B N (V m ) is a term representing NMDAR voltage-dependent Mg 2+ unblock (fit from [4]), given as The calcium-activated SK-current is given by, where c is the local Ca 2+ concentration at the SK-channel, τ s is the SK-activation time constant, K m is the half-activation parameter and n is the Hill coefficient [5].
Ca 2+ fluxes Ca 2+ fluxes are calculated from the Ca 2+ component of the NMDAR current and the VGCC currents. The Ca 2+ component of the NMDAR current is given by where P f is the fractional contribution of Ca 2+ to the NMDAR current at −60mV, and E Ca is the Ca 2+ reversal potential. The additional correction factor of 1/3 is applied to P f , due to the difference in reversal potentials for the NMDAR current Ca 2+ component and total NMDAR current.
VGCC Ca 2+ currents are modeled as in [6] where the activation and inactivation gating variables take the form where σ ∈ {m CaT , h CaT , m CaH , h Ca h }. The steady-state functions, σ ∞ , are given by All Ca 2+ currents were converted to Ca 2+ fluxes where υ ∈ {N, A, Ca T , Ca L }, F is Faraday's constant, and Γ υ is the surface-area on the spine boundary representing the channel cluster.

Ca 2+ extrusion
The various Ca 2+ extrusion mechanisms in the spine were modeled as a single linear term in Ca 2+ concentration, c. The extrusion rate, γ, is constant and was estimated using Equation 4 in [7], and Ca 2+ imaging data from [8] and [9].

Ca 2+ buffering
For the simplest case (those simulations not involving the complex Ca 2+ /calmodulin model), the binding of Ca 2+ to buffer species, B, was modeled simply as with apparent forward and backward binding rates, k + and k − . Inclusion of the Ca 2+ -extrusion term and assuming an unchanged D B on binding of the buffer to the much smaller Ca 2+ ion and uniform initial buffer concentration leads to the following reaction-diffusion system where c, b and u are the Ca 2+ , mobile buffer, and EFB concentrations respectively. EFB concentrations were calculated using the following relationship, which is valid when the resting Ca 2+ concentration is taken as zero, as described in [10] where κ u is the EFB binding ratio and K u is the EFB affinity for Ca 2+ .

Cooperative binding of Ca 2+ to calmodulin
We used a detailed model of cooperative binding of Ca 2+ to calmodulin [11]. The reaction network as described in [11] is shown in Supplementary Figure 1. Ca 2+ /calmodulin binding-scheme reaction-rates are the same as in [11]. ** Parameter was tuned to obtain best match to experimental data from the referenced source.