The proposed model is an extension of the previous one, which predicted the formation of local Ca2+ microdomains in PsCs due to the reversal of the Na+/Ca2+ exchanger (NCX) during neuronal activity 14. The present model is a five-compartment conductance-based model consisting of (i) a presynaptic axon terminal (Pre), (ii) a postsynaptic dendritic spine (Post), (iii) a local extracellular space (ECS), (iv) a PsC and (v) a global extracellular space (GECS). Neuronal and astrocytic compartments have various ionic channels and transporters dwelling in their respective membranes, which are modelled using well-established conductance-based equations. The compartments and ionic channels used in the model are shown in Fig. 1. The compartments are assumed to have uniformly distributed concentrations of three ionic species: Na+, K+ and Ca2+. The model consists of six key variables which control the state of the synapse: the concentrations of Na+, K+ and Ca2+ in the PsC and ECS compartments. The GECS is assumed to have a much larger volume than the ECS and therefore the concentrations of all ions are held constant in the GECS. Concentration changes are described in terms of transmembrane currents produced by the electrogenic ion channels interfacing with each compartment. Concentration changes take the general form:

$$\begin{array}{c}\frac{d{\left[x\right]}_{y}}{dt}=-\left(\frac{{I}_{x}}{{z}_{x}F Vo{l}_{y}}\right)\left(1\right)\end{array}$$

where *x* is the ion under consideration (Na+, K+ and Ca2+), *y* is the compartment under consideration (Pre, Post, PsC and ECS), *I**x* is the total current of ion *x* entering/leaving the compartment *y*, *z**x* is the ionic valence, *F* is Faraday’s constant and *Vol**y* is the volume of compartment *y*. The equations and parameters for the currents of the channels shown in Fig. 1 are given in Supplementary Tables 1 and 2. The membrane potential of each neuronal cell is described using a standard Hodgkin-Huxley type ohmic equation:

$$\begin{array}{c}{C}_{m}\frac{d{V}_{y}}{dt}={I}_{ext}-I\left(2\right)\end{array}$$

where *C**m* is the membrane capacitance, *V**y* is the membrane potential of compartment *y*, *I**ext* is an externally applied current and *I* is the total transmembrane current. Our previous model used a single-compartment synapse 14, whereas here, we split the synapse into two separate compartments: Pre and Post. The Pre compartment has voltage-gated Ca2+ channels (VGCCs) necessary for exocytosis of Glu and the Post compartment has Glu receptors necessary for successful synaptic transmission: N-methyl-D-aspartate receptor (NMDA-R) and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPA-R).

Note that Table 1 in the appendix gives details of all models used in this work. However, we wish to highlight a simple transporter model based on Michaelis-Menten (MM) kinetics. The model uses extracellular Glu as the enzyme, similar to ATP transporter models. Using MM kinetics allows us to describe the EAAT with parameters that represent physiological conditions. Therefore, the EAAT is given:

$$\begin{array}{c}{I}_{eaat,psc}={V}_{eaat}ef{f}_{eaat}F\left(\frac{{\left[Glu\right]}_{ecs}}{{K}_{eaat}+{\left[Glu\right]}_{ecs}}\right)\left(3\right)\end{array}$$

where *V**eaat* is the maximal EAAT velocity, *eff**eaat* is the average EAAT efficiency, *K**eaat* is the EAAT Glu affinity and [Glu]*ecs* is the Glu concentration in the cleft.

In the model, Glu is released from the presynaptic ’active zone’ due to Ca2+ influx through VGCCs. Glu released during periods of neuronal activity activates postsynaptic Glu receptors and is cleared through diffusion and uptake by electrogenic EAATs on the surrounding PsC membrane. The rate of transmembrane Glu translocation by EAATs is relatively slow, approaching about 30 molecules of Glu per second 15,16. The binding of Glu to the transporters (Km ≈ 20 µM), is much faster, with a binding rate up to 107 M− 1s− 1 17, and hence Glu transporters concentrated at the PsC provide for an almost instant buffering of Glu. The higher the density of transporters, the higher their buffering capacity 18. It has been estimated that there are approximately 8,000–10,000 EAATs per µm2 concentrated at the PsC 19. The efficacy of EAAT Glu transport is about 50% 18, meaning that approximately half of the Glu molecules dissociate from transporters and bind again to nearby receptors or other EAATs, which may affect the kinetics of Glu presence in the synaptic cleft. In this study, we fix the density of EAATs at 10,000 µm− 2, the EAAT efficacy is 50% and, using Eq. 3 and based on the figures given, we calculate the time it takes the EAATs to clear 1 mM of Glu to be ∼33 ms. For each transport cycle, the EAAT co-transports 1Glu:3Na+:1H+ and counter-transports 1K+, generating a PsC Na+ influx and K+ efflux (it should be noted that the proposed model does not account for the H+ fluxes or Cl− which is thermodynamically uncoupled from Glu transport). The background activity of the NKA increases due to the intracellular Na+ and extracellular K+ concentrations changing from the respective transmembrane fluxes. Separate to K+ release through the EAAT, astrocytic K+ efflux is achieved through Kir4.1 channels densely populating perisynaptic membranes. The main synaptic K+ efflux pathway in the model is the current generated through the postsynaptic NMDA-R. Hodgkin-Huxley type voltage-gated K+ and Na+ channels are included for the generation of the presynaptic action potential. However, we assume the presynaptic voltage-gated K+ and Na+ channels do not interact with the synapse ’active zone’ 20 therefore we exclude the contribution of these channels to the respective ionic concentration in the ECS.

To simulate the movement of the PsC during synaptic activation we carried out simulations for: A) when the PsC is in intimate contact with the synapse and the ECS contains the cleft region only, B) when a space exists between the PsC and the synapse such that the ECS volume is seven times that of the cleft and C) when a space exists such that the ECS volume is approximately forty-nine times that of the cleft; Fig. 2 shows a schematic of the three levels of PsC coverage. The PsC and neuronal compartment volumes and surface areas are kept constant throughout all simulations.

The model was implemented using MATLAB 2019b (64-bit Windows version) by Mathworks. All simulation results presented in this section use a forward Euler numerical integration scheme with a fixed time step of t = 10 µs and all model simulations were carried out for 100 seconds. All transmembrane currents are given in femtoamps (fA), positive currents denote ions leaving the cellular compartment (negative ionic flux) and negative currents denote ions entering the cellular compartment (positive ionic flux).