astrocyte_lr_1994 – An astrocyte model based on Li & Rinzel (1994)
==================================================================
Description
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``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 (:math:`\mathrm{[IP3]}`), the calcium
concentration in the astrocytic cytosol (:math:`\mathrm{[Ca^{2+}]}`), and the
fraction of IP3 receptors on the astrocytic endoplasmic reticulum (ER) that are
not yet inactivated by calcium (:math:`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:
.. math::
\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, :math:`\mathrm{[IP3]}` decays to its baseline value
(:math:`[\mathrm{IP3}]_0`) with the time constant (:math:`\tau_\mathrm{IP3}`). Each time when an
astrocyte receives an excitatory synaptic input, it triggers an instantaneous
increase of :math:`\mathrm{[IP3]}`. In this implementation, the inputs are spike events
sent from neurons or generators. The summed synaptic weight the astrocyte receives at time
:math:`t` is given by :math:`J_\mathrm{syn}(t)`. The parameter
:math:`\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:
.. math::
\frac{d[\mathrm{Ca^{2+}}]}{dt} =
J_\mathrm{channel} - J_\mathrm{pump} + J_\mathrm{leak} + J_\mathrm{noise}
Here, :math:`\mathrm{[Ca^{2+}]}` is the cytosolic calcium concentration, and
:math:`J_\mathrm{noise}` is the current input. :math:`J_\mathrm{channel}`,
:math:`J_\mathrm{pump}`, :math:`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]:
.. math::
I_\mathrm{SIC} = \mathrm{SIC_{scale}} \cdot \mathrm{H}\left(\mathrm{ln}(y)\right) \cdot \mathrm{ln}(y)
where
.. math::
y = \left( \mathrm{[Ca^{2+}]} - \mathrm{SIC_{th}} \right)/\mathrm{nM}
When the cytosolic calcium concentration of the astrocyte exceeds the threshold
value (:math:`\mathrm{SIC_{th}}`), a SIC output (:math:`I_\mathrm{SIC}`) is
generated. This thresholding is modeled as a Heaviside function
(:math:`\mathrm{H(\cdot)}`). In this implementation, the SIC threshold
:math:`\mathrm{SIC_{th}}` as well as the scaling constant
:math:`\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 <../model_details/astrocyte_model_implementation.ipynb>`_ notebook.
See also [1]_, [2]_, [3]_.
Parameters
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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] 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] 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] 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
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SICEvent
Receives
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SpikeEvent, DataLoggingRequest
See also
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:doc:`Astrocyte `
Examples using this model
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.. listexamples:: astrocyte_lr_1994