Fig 1.
Compartments, ion channels and transporters included in the modelling of the glutamatergic synapse.
Shown are the three main components representing a presynaptic neuron, an astrocyte and the extracellular space (ECS). Each of these compartments also contains a synaptic compartment as indicated by the different shading and the additional box (presynaptic terminal, perisynaptic process and synaptic cleft, respectively). The largest ATP consumption in the presynaptic neuron and the astrocyte is by the Na+/K+-ATPase (NKA). At the presynaptic terminal, ATP is also needed to energize glutamate uptake into vesicles. The key transporters at the cleft are the Na+/Ca2+-exchanger (NCX) and the Excitatory Acid Amino Transporter (EAAT). NKCC1: Na+-K+-Cl−-cotransporter. KCC: K+-Cl−-cotransporter. Kir4.1: K+ inward rectifier channel 4.1.
Fig 2.
Glutamate recycling scheme, inspired by combining vesicle-based models from [42] and [43].
(Left) Closer view of the model scheme at the synapse and (Right) the glutamate recycling scheme. Inactive neuronal intracellular glutamate (I) moves to the depot (D) from where it is packed into vesicles which pass through five stages (N, R, R1,2,3) before they are released into the synaptic cleft (F). These stages have fast time-constants that depend on intracellular Ca2+ concentration. The stages Ri correspond to vesicles that are bound by i Ca2+ ions. The time-constants change when there is influx of Ca2+ in the presynaptic terminal in response to membrane depolarization. Released glutamate in the cleft can be taken up by astrocytes or back to neurons using excitatory amino acid transporters (EAATs) or leak channels, thereby recycling the released neurotransmitter.
Table 1.
Notation used in the model equations.
Fig 3.
Model calibrations reproduce experimental data.
(A) Plot of versus time t. ED begins at t = tstart min. and ends at t = tend min. while being reduced to a minimum of Pmin. (B,C) Experimental traces from [8] (left) and the corresponding model simulations (right) with (B) showing intracellular sodium for neurons and astrocytes and (C) showing extracellular sodium and potassium. Empirically adjusted parameters: Pmin and
(initial extracellular volume fraction) were chosen by fitting model dynamics qualitatively to Na+ and K+ concentration time-traces. Here, Pmin = 50% and
. The difference in Na+ increase between neurons and astrocytes is attributed to the presence of fast Na+ influx through voltage-gated Na+ channels in neurons, which are lacking in astrocytes. Please note that scaling axis in panels B and C may slightly differ between experiments and simulations for optimal display purposes.
Table 2.
Parameter values for all simulations performed.
Units are presented in the same manner as they are implemented in the Python code.
Fig 4.
Shown are the time courses of the membrane potential, sodium and potassium concentrations, cell volume and glutamate in response to a current pulse (25 pA, 10 s, black trace) with (A) and without (B) a functional astrocyte.
(A) In response to the pulse, there is a burst of action potentials and return to baseline of all quantities. (B) Without a functional astrocyte, the neuron depolarizes after the burst, and remains in this state, even if the astrocyte function is restored. Here, we plot neuronal (blue), astrocyte (orange) and extracellular (green) traces against time for several quantities. The initial extracellular volume ratio is αe = 20%. The shaded red area corresponds to periods during which ion transport across the astrocytic plasma membrane is blocked.
Fig 5.
ED for 5 minutes (A) and 15 minutes (B) demonstrates the existence of two stable states: 1) before ED (baseline resting state) and 2) after prolonged ED of 15 minutes (stable depolarized state).
Here, we plot neuronal (blue), astrocyte (orange) and extracellular (green) traces against time for several quantities. The initial extracellular volume ratio αe = 80% and minimal energy available Pmin = 50%. Shaded grey areas correspond to the period where Na+/K+-ATPase (NKA) activity is gradually reduced to Pmin and restored to baseline, identical to Fig 3A.
Fig 6.
Differential sensitivity of ED to initial extracellular volume ratio αe (A) and to baseline Na+/K+-ATPase strength factor Pscale (B).
(A) We deprive the neuron and astrocyte of energy for 5 minutes before restoring it to baseline and report the relative volume 30 minutes after restoration. We show two examples, (A.1) for large ECS (αe = 80%) and (A.2) for realistic ECS (αe = 20%). Here, we plot neuronal (blue), astrocyte (orange) and extracellular (green) traces against time for compartmental volume change. (B) We deprive the neuron and astrocyte of energy for 15 minutes before restoring it to baseline, for two different values of Pscale and αe. We show neuronal and astrocyte membrane potentials against time. The grey area in (B.1–4) illustrates the period of ED. The table in the middle indicates whether the system recovers (green) post energy restoration or not (red).
Fig 7.
Tipping in a bistable regime (A) and the change in tipping behavior by introducing pharmacological blockers (B).
In (A.1a and A.2a), we plot bifurcation diagrams with respect to Pmin for αe = 20% (realistic ECS) and αe = 80% (large ECS). Red curves are pathological branches, and blue curves are physiological branches. Dashed lines represent unstable parts. The only two relevant local bifurcations are limit point (star) and Hopf (inverted triangle). The inset shows two additional bifurcations, a pitchfork (dot) and a Hopf. We show two simulations (A.1b and A.2b), short ED (5 minutes, cyan curve) and long ED (15 minutes, pink curve), both for Pmin = 50%. In (B), we block different pathways during ED (green area, B.1, B.2 and B.3) and after restoration (green area, B.4). For (B.1–3), energy is deprived for 5 minutes for parameters αe = 80% and Pmin = 50%. In (B.4), energy is deprived for 15 minutes for parameters αe = 80% and Pmin = 50%.
Fig 8.
Experimental findings from Brisson and Andrew [32] (reproduced with permission) and our model simulations.
(A) (Left) Membrane depolarization of a pyramidal neuron during 10 minutes of oxygen glucose deprivation (OGD), that persists after restoring energy. (Right) Model simulations on the right show the neuronal (blue) and astrocyte membrane potentials. ED (OGD) is introduced for 15 minutes (red line). Here, αe = 80% and Pmin = 0%. The dynamics are faithfully reproduced, including anoxic oscillations at the initial phase of depolarization. (B) (Left) Membrane potential of a magnocellular neuroendocrine cell, showing a similar depolarization during 15 minutes oxygen glucose deprivation and full recovery after this period. Both at the start of the depolarization and during recovery action potentials are generated. (Right) Model simulations. ED (OGD) is introduced for 15 minutes (black line). Here, αe = 80%, Pmin = 0% and Pscale = 2. With these parameter settings, the membrane potential recovers to baseline conditions after the period with ED. Note that during recovery, the experimentally observed oscillations are also faithfully simulated.
Fig 9.
Neuronal stimulation upon recovery from ED shows different glutamate transients as compared to neuronal stimulation in a pathological state.
Recovery from ED is achieved by blocking neuronal voltage-gated Na+ channels (A) and blocking neuronal voltge-gated K+ channels (B). Glutamate in the cleft (green trace) and neuronal membrane potential (blue trace) are shown, in response to neuronal excitation in physiological and pathological conditions. First, ED is simulated between t = 5 and t = 20 minutes (Pmin = 50%, αe = 80%). Then, neurons are subjected to 25 pA square wave input for 10 seconds, as indicated by the black trace. The system is then brought back to the physiological state by blocking voltage-gated Na+ channels (shaded green area). After a little more than an hour, the neurons are subjected again, to 25 pA square wave input for 10 seconds.
Table 3.
Common model parameters along with sources.
Units are presented in the same manner as they are implemented in the Python code. All adjusted parameters are in the same order of magnitude as their original counterparts.
Table 4.
Model parameters for the neuronal compartment, along with sources.
Units are presented in the same manner as they are implemented in the Python code. All adjusted parameters are in the same order of magnitude as their original counterparts.
Table 5.
Model parameters for the astrocyte compartment, along with sources.
Units are presented in the same manner as they are implemented in the Python code.
Table 6.
Model parameters for glutamate recycling, along with sources.
Units are presented in the same manner as they are implemented in the Python code.
Table 7.
Initial values for the various states in the model.
These values correspond to ‘baseline’ conditions, and are used to estimate unknown parameters. Units are presented in the same manner as they are implemented in the Python code.
Table 8.
Parameters estimated from baseline conditions.
Units are presented in the same manner as they are implemented in the Python code. See section ‘Estimating parameters’ for derivation.