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:

\[\frac{d[\mathrm{IP3}]}{dt} = \frac{[\mathrm{IP3}]_0 - [\mathrm{IP3}]}{\tau_\mathrm{IP3}} + \Delta_\mathrm{IP3} \cdot J_\mathrm{syn}(t)\]

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:

\[\frac{d[\mathrm{Ca^{2+}}]}{dt} = J_\mathrm{channel} - J_\mathrm{pump} + J_\mathrm{leak} + J_\mathrm{noise}\]

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]:

\[I_\mathrm{SIC} = \mathrm{SIC_{scale}} \cdot \mathrm{H}\left(\mathrm{ln}(y)\right) \cdot \mathrm{ln}(y)\]

where

\[y = \left( \mathrm{[Ca^{2+}]} - \mathrm{SIC_{th}} \right)/\mathrm{nM}\]

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.

See also [1], [2], [3].

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

[1] (1,2)

Li, Y. X., & Rinzel, J. (1994). Equations for InsP3 receptor-mediated [Ca2+]i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. Journal of theoretical Biology, 166(4), 461-473. DOI: https://doi.org/10.1006/jtbi.1994.1041

[2] (1,2)

De Young, G. W., & Keizer, J. (1992). A single-pool inositol 1,4,5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration. Proceedings of the National Academy of Sciences, 89(20), 9895-9899. DOI: https://doi.org/10.1073/pnas.89.20.9895

[3] (1,2,3,4,5)

Nadkarni, S., & Jung, P. (2003). Spontaneous oscillations of dressed neurons: a new mechanism for epilepsy?. Physical review letters, 91(26), 268101. DOI: https://doi.org/10.1103/PhysRevLett.91.268101

Sends

SICEvent

Receives

SpikeEvent, DataLoggingRequest

See also

Astrocyte

Examples using this model

• A model using a single astrocyte with calcium dynamics

• A small neuron-astrocyte network

• A tripartite interaction between two neurons and one astrocyte

• Random balanced network with astrocytes