A Ca-Based Computational Model for NMDA Receptor-Dependent Synaptic Plasticity at Individual Post-Synaptic Spines in the Hippocampus

Abstract

Associative synaptic plasticity is synapse specific and requires coincident activity in pre-synaptic and post-synaptic neurons to activate NMDA receptors (NMDARs). The resultant Ca(2+) influx is the critical trigger for the induction of synaptic plasticity. Given its centrality for the induction of synaptic plasticity, a model for NMDAR activation incorporating the timing of pre-synaptic glutamate release and post-synaptic depolarization by back-propagating action potentials could potentially predict the pre- and post-synaptic spike patterns required to induce synaptic plasticity. We have developed such a model by incorporating currently available data on the timecourse and amplitude of the post-synaptic membrane potential within individual spines. We couple this with data on the kinetics of synaptic NMDARs and then use the model to predict the continuous spine [Ca(2+)] in response to regular or irregular pre- and post-synaptic spike patterns. We then incorporate experimental data from synaptic plasticity induction protocols by regular activity patterns to couple the predicted local peak [Ca(2+)] to changes in synaptic strength. We find that our model accurately describes [Ca(2+)] in dendritic spines resulting from NMDAR activation during pre-synaptic and post-synaptic activity when compared to previous experimental observations. The model also replicates the experimentally determined plasticity outcome of regular and irregular spike patterns when applied to a single synapse. This model could therefore be used to predict the induction of synaptic plasticity under a variety of experimental conditions and spike patterns.

Keywords: NMDA receptor; dendritic spines; hippocampus; spike timing-dependent plasticity; synaptic plasticity.

Figure 1

Calculating [Ca²⁺] in dendritic spines during continuous pre- and post-synaptic activity. The model initially calculates the membrane potential during continuous activity by summating the membrane potential changes due to EPSPs and BPAPs from a resting membrane potential of −65 mV. The Ca²⁺ current passing through synaptic NMDARs is then calculated from the membrane potential and glutamate binding kinetics. Finally spine [Ca²⁺] is calculated from Ca²⁺ buffering and diffusion kinetics. Left hand panels show the post-synaptic responses to an epoch of place cell activity spanning 3500 ms. Right hand panels show a 200 ms excerpt from this epoch.

Figure 2

Comparison of predicted [Ca²⁺] dynamics in dendritic spines and in the soma. [Ca²⁺] profiles in response to a 10 mV EPSP at the spine (A) or a 1 mV EPSP at the soma (B) on their own (gray) or in combination with a BPAP (black) of amplitude 60 mV at the spine (A) or 100 mV at the soma (B). The delay between EPSP and BPAP initiation is 10 ms.

Figure 3

[Ca²⁺] dynamics in response to paired pre- and post-synaptic spikes. (A) The model calculates [Ca²⁺] within a spine from the membrane potential resulting from a pair of pre- and post-synaptic spikes. Gray line shows EPSP in the absence of BPAP. Varying Δt shows that [Ca²⁺]_max is greatest when 0 ≤ Δt ≤ 30 ms for 10 mV (B) or 20 mV (C) EPSPs. (D) The frequency of spike pairings given at Δt = 10 ms determines [Ca²⁺]_max.

Figure 4

[Ca²⁺] dynamics in response to triplets of one pre- and two post-synaptic spikes. (A) The model calculates [Ca²⁺] within a spine from the membrane potential resulting from a triplet of pre- and post-synaptic spikes. Gray line shows EPSP in the absence of BPAP. (B). Varying Δt shows that [Ca²⁺]_max is greatest when 0 ≤ Δt ≤ 30 ms for 10 or 20 mV EPSPs for Δs = 10 ms. (C) Varying Δs shows that [Ca²⁺]_max decreases as Δs increases for 10 or 20 mV EPSPs and Δt = 10 ms. (D) The frequency of spike pairings given at Δt = 10 ms and Δs = 10 ms determines [Ca²⁺]_max.

Figure 5

Theta burst pairing produces large spine [Ca²⁺]. The model calculates [Ca²⁺] within a spine from the membrane potential resulting from coincident theta burst stimulation of pre- and post-synaptic neurons (black) or only pre-synaptic neuron (gray).

Figure 6

Post-synaptic voltage clamp paired with pre-synaptic stimulation determines spine [Ca²⁺]. The model predicts that voltage clamp of the post-synaptic membrane potential at −40 mV produces a much smaller spine [Ca²⁺] than 0 mV when paired with a single pre-synaptic stimulation.

Figure 7

Spine [Ca²⁺] determines the direction and magnitude of synaptic weight change. (A). The Ω−function describes the relationship between peak spine [Ca²⁺] and synaptic weight change. Symbols represent the peak [Ca²⁺] produced by a single application of the plasticity induction protocols shown in Figures 3–6 and indicate the resulting predicted synaptic weight change. (B) The η-function describes the learning rate for synaptic weight change as a function of peak spine [Ca²⁺].

Figure 8

Example of predicted synaptic weight change during overlapping place cell activity. The model calculates spine [Ca²⁺] during a ∼16-min period of activity from two place cells (1_A and 1_B) with overlapping place fields. The synaptic weight change is then calculated from the peak spine [Ca²⁺] and shows a robust, rapidly developing potentiation.

Figure 9

Predicted synaptic weight changes for overlapping and non-overlapping place cell activity. Calculated synaptic weight changes for four pairs of overlapping place cells 2_A, 2_B (A), 2_C, 2_D (B), 3_A, 3_B (C), and 4_A, 4_B (D) as well as one pair of non-overlapping place cells 1_A, 1_C (E) and one pair of adjacent place cells 2_E, 2_D (F).

Figure 10

Predicted synaptic weight changes for place cell activity with specific spike patterns. Calculated synaptic weight changes for a pair of overlapping place cells with an asymmetric cross-correlation 1_B, 1_A (A) and a pair of place cells where all spike intervals with positive Δt less than 100 ms have been removed 1_A, 1_B (B). (C) A comparison between the induced plasticity predicted by the model and the observed plasticity from experimental data (Isaac et al., 2009).