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A mathematical model for astrocytes mediated LTP at single hippocampal synapses

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Abstract

Many contemporary studies have shown that astrocytes play a significant role in modulating both short and long form of synaptic plasticity. There are very few experimental models which elucidate the role of astrocyte over Long-term Potentiation (LTP). Recently, Perea and Araque (Science 317:1083–1086, 2007) demonstrated a role of astrocytes in induction of LTP at single hippocampal synapses. They suggested a purely pre-synaptic basis for induction of this N-methyl-D-Aspartate (NMDA) Receptor-independent LTP. Also, the mechanisms underlying this pre-synaptic induction were not investigated. Here, in this article, we propose a mathematical model for astrocyte modulated LTP which successfully imitates the experimental findings of Perea and Araque (Science 317:1083–1086, 2007). Our study suggests the role of retrograde messengers, possibly Nitric Oxide (NO), for this pre-synaptically modulated LTP.

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Acknowledgments

The work has been supported by the Department of Science and Technology, Government of India, grant no. SR/CSI/08/2009. Helpful discussions and suggestions from Prof. Alfonso Araque, Instituto Cajal, Spain is being thankfully acknowledged.

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Correspondence to Kaushik Majumdar.

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Appendix A

Appendix A

The idea of the astrocytic feedback into the pre-synaptic terminal is not new as it has been previously modeled by Nadkarni et al. (2008). However our model is new and more challenging in the sense that it is biologically more detailed than Nadkarni et al. (2008). For example, Nadkarni et al. (2008) modeled a CA3–CA1 pyramidal cell (CA3–CA1) synapse as modulated by an astrocyte. It needs to be pointed out that the parameter values chosen by them for bouton calcium in response to an action potential (AP) are biologically not valid at the CA3–CA1 synapses. They assume that the intracellular calcium concentration rises instantly to 300 μM (and falls back to the resting values after 2 ms) in response to an AP. If we assume such high concentration then it implies that nearly \( \frac{{0.13 \times 300 \times {{10}^{ - 6}} \times 6.023 \times {{10}^{23}}}}{{{{10}^{15}}}} \approx 23490 \) ions (for an average pyramidal cell bouton of volume 0.13 μm3, Koester and Sakmann 2000) are free per AP. However the total calcium charge entering into a hippocampal pyramidal cell bouton during an AP is less than 1 fC (femto Coulomb) (Koester and Sakmann 2000). It simply implies that the total calcium ions entering a hippocampal pyramidal bouton during an AP is less than \( \frac{{1 \times {{10}^{ - 15}}}}{{2 \times 1.6 \times {{10}^{ - 19}}}} - 3125 \) ions which is considerably much less than 23490 ions as discussed before. In our case, calcium entering through an AP is around 5 μM i.e. \( \frac{{0.13 \times 5 \times {{10}^{ - 6}} \times 6.023 \times {{10}^{23}}}}{{{{10}^{15}}}} \approx 392 \) ions which is well under 3125 ions (maximum number of calcium ions entering into a hippocampal pyramidal bouton).

Furthermore, Weber et al. (2010) concluded that the neurons with extracellular calcium concentration of 2 mM are unlikely to have calcium sensor affinity as high as 100 μM which makes the choice of Bertram model unfavorable (since its four sensors or sites have affinities 108 nM, 400 nM, 200 μM and 1334 μM) at least for the modeling of the pre-synaptic bouton neurotransmitter release. Instead of high calcium sensor affinity Weber et al. (2010) suggested 10 μM as the best estimate for calcium sensor affinity which has been used in our model.

The astrocytic calcium dynamics used in Nadkarni et al. (2008) and in our model may seem identical in broad sense (since both of them have an agonist-dependent inositol triphosphate (IP3) production term, agonist-independent IP3 production term and IP3 degradation term), but they are not. For example, the calcium-dependent term used in their model is based on the classic De Young and Keizer (1992) model which talks about phospholipase C (PLC) (which is activated by calcium) dependent IP3 production is based on data from the experiments over the WRK1 cells (Mouillac et al. 1990), liver cells (Taylor and Exton 1987) etc. On the other hand we make use of the G-ChI model (De Pitta et al., 2009), which has a more detailed IP3 degradation term (it incorporates inositol polyphosphate 5-phosphatase (IP-5P) and IP3 3-kinase (IP3-3 K) based IP3 degradation), agonist-dependent term (it incorporates Ca2+/protein kinase C (PKC)-dependent inhibitory factor over agonist-dependent IP3 production term) and agonist-independent IP3 production term (it incorporates PLCδ-dependent IP3 production term) based on an astrocyte specific experiment.

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Tewari, S., Majumdar, K. A mathematical model for astrocytes mediated LTP at single hippocampal synapses. J Comput Neurosci 33, 341–370 (2012). https://doi.org/10.1007/s10827-012-0389-5

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