astrocyte_lr_1994 – An astrocyte model based on Li & Rinzel (1994)¶
Description¶
astrocyte_lr_1994
is a model of astrocytic calcium dynamics. The model was
first proposed by Li & Rinzel (1994) [1] and it is based on earlier work of De
Young & Kaiser (1992) [2]. The input and output of the model are implemented
according to Nadkarni & Jung (2003) [3].
The model has three dynamic state variables: the concentration of inositol 1,4,5-trisphosphate in the astrocyte (\(\mathrm{[IP3]}\)), the calcium concentration in the astrocytic cytosol (\(\mathrm{[Ca^{2+}]}\)), and the fraction of IP3 receptors on the astrocytic endoplasmic reticulum (ER) that are not yet inactivated by calcium (\(h_\mathrm{IP3R}\)).
In this model, excitatory synaptic inputs to the astrocyte trigger IP3 generation and change in calcium dynamics in the astrocyte. This might induce an astrocytic output in the form of a slow inward current (SIC), which is dependent on its calcium dynamics, to its target neurons. The input and output are based on the equations in [3] but with adaptations, as described in the following.
Spike input¶
The astrocyte receives inputs from excitatory (glutamatergic) synapses. The synaptic inputs directly affect IP3 concentration according to the following equation:
In the absence of inputs, \(\mathrm{[IP3]}\) decays to its baseline value (\([\mathrm{IP3}]_0\)) with the time constant (\(\tau_\mathrm{IP3}\)). Each time when an astrocyte receives an excitatory synaptic input, it triggers an instantaneous increase of \(\mathrm{[IP3]}\). In this implementation, the inputs are spike events sent from neurons or generators. The summed synaptic weight the astrocyte receives at time \(t\) is given by \(J_\mathrm{syn}(t)\). The parameter \(\Delta_\mathrm{IP3}\) scales the impact of synaptic inputs on the IP3 dynamics.
Calcium current input¶
In this implementation, a current input to the astrocyte is directly added to its cytosolic calcium concentration. Generators that send out currents can be connected to astrocytes to directly generate fluctuations in cytosolic calcium:
Here, \(\mathrm{[Ca^{2+}]}\) is the cytosolic calcium concentration, and \(J_\mathrm{noise}\) is the current input. \(J_\mathrm{channel}\), \(J_\mathrm{pump}\), \(J_\mathrm{leak}\) are the calcium fluxes defined as in [3].
Output¶
If the astrocyte receives excitatory synaptic inputs, it might output SIC to its target neurons. This current depends on the cytosolic calcium concentration. This dependency is modeled according to the expressions first proposed in [3]:
where
When the cytosolic calcium concentration of the astrocyte exceeds the threshold value (\(\mathrm{SIC_{th}}\)), a SIC output (\(I_\mathrm{SIC}\)) is generated. This thresholding is modeled as a Heaviside function (\(\mathrm{H(\cdot)}\)). In this implementation, the SIC threshold \(\mathrm{SIC_{th}}\) as well as the scaling constant \(\mathrm{SIC_{scale}}\) are treated as model parameters that can be set together with other parameters. Nadkarni & Jung (2003) [3] proposed values for these parameters by fitting the equation for SIC to an experimental data set.
The output is implemented as SICEvent sent from the astrocyte to its target
neurons through the sic_connection
.
For the reference implementation of this model, see the astrocyte_model_implementation notebook.
Parameters¶
The following parameters can be set in the status dictionary.
Dynamic state variables |
||
IP3 |
µM |
Inositol 1,4,5-trisphosphate concentration in the astrocytic cytosol |
Ca_astro |
µM |
Calcium concentration in the astrocytic cytosol |
h_IP3R |
unitless |
Fraction of IP3 receptors on the astrocytic ER that are not yet inactivated by calcium |
Parameters |
||
Ca_tot |
µM |
Total free astrocytic calcium concentration in terms of cytosolic volume |
IP3_0 |
µM |
Baseline value of astrocytic IP3 concentration |
Kd_IP3_1 |
µM |
First astrocytic IP3R dissociation constant of IP3 |
Kd_IP3_2 |
µM |
Second astrocytic IP3R dissociation constant of IP3 |
Kd_act |
µM |
Astrocytic IP3R dissociation constant of calcium (activation) |
Kd_inh |
µM |
Astrocytic IP3R dissociation constant of calcium (inhibition) |
Km_SERCA |
µM |
Half-activation constant of astrocytic SERCA pump |
SIC_scale |
unitless |
Parameter determining the scale of astrocytic SIC output |
SIC_th |
µM |
Threshold that determines the minimal level of astrocytic cytosolic calcium sufficient to induce SIC |
delta_IP3 |
µM |
Parameter determining the increase in astrocytic IP3 concentration induced by synaptic input |
k_IP3R |
1/(µM*ms) |
Astrocytic IP3R binding constant for calcium inhibition |
rate_IP3R |
1/ms |
Maximum rate of calcium release via astrocytic IP3R |
rate_L |
1/ms |
Rate constant of calcium leak from astrocytic ER to cytosol |
rate_SERCA |
µM/ms |
Maximum rate of calcium uptake by astrocytic SERCA pump |
ratio_ER_cyt |
unitless |
Ratio between astrocytic ER and cytosol volumes |
tau_IP3 |
ms |
Time constant of the exponential decay of astrocytic IP3 |
References¶
Sends¶
SICEvent
Receives¶
SpikeEvent, DataLoggingRequest