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 (: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 ++++++++++ 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 +++++ SICEvent Receives ++++++++ SpikeEvent, DataLoggingRequest See also ++++++++ :doc:`Astrocyte ` Examples using this model +++++++++++++++++++++++++ .. listexamples:: astrocyte_lr_1994