Underlying Mechanisms of Cooperativity, Input Specificity, and Associativity of Long-Term Potentiation Through a Positive Feedback of Local Protein Synthesis

Long-term potentiation (LTP) is a specific form of activity-dependent synaptic plasticity that is a leading mechanism of learning and memory in mammals. The properties of cooperativity, input specificity, and associativity are essential for LTP; however, the underlying mechanisms are unclear. Here, based on experimentally observed phenomena, we introduce a computational model of synaptic plasticity in a pyramidal cell to explore the mechanisms responsible for the cooperativity, input specificity, and associativity of LTP. The model is based on molecular processes involved in synaptic plasticity and integrates gene expression involved in the regulation of neuronal activity. In the model, we introduce a local positive feedback loop of protein synthesis at each synapse, which is essential for bimodal response and synapse specificity. Bifurcation analysis of the local positive feedback loop of brain-derived neurotrophic factor (BDNF) signaling illustrates the existence of bistability, which is the basis of LTP induction. The local bifurcation diagram provides guidance for the realization of LTP, and the projection of whole system trajectories onto the two-parameter bifurcation diagram confirms the predictions obtained from bifurcation analysis. Moreover, model analysis shows that pre- and postsynaptic components are required to achieve the three properties of LTP. This study provides insights into the mechanisms underlying the cooperativity, input specificity, and associativity of LTP, and the further construction of neural networks for learning and memory.

Upon stimulation, the presynaptic neuron releases neurotransmitter to trigger the postsynaptic membrane potential.
First, we referred to the discrete vesicle release model for the neurotransmitter dynamics in each synapse (Tsodyks and Markram, 1997;Nadkarni et al., 2008).
There are n (j = 1, 2, · · · , n) synapses in the dendrite (Fig. 1A). For each synapse, the changes of glutamate concentration [G] j in the cleft is given by the equation where τ in is the time constant of inactivation. Here, the delta function δ(t − t i,j ) represents the events of stimulated vesicle release at each synapse at the discrete time series t i,j . The time series t i,j is determined by the protocol of the stimulus to the specific synapse. In this study, we divided the synapses into two pathways, which include n 1 (the pathway P1, synapses j = 1, · · · , n 1 ) and n 2 (the pathway P2, synapses j = n 1 + 1, · · · , n 1 + n 2 = n) synapses, respectively, and different stimulus protocols are applied to the two pathways. In each synapse, the neurotransmitter content of a vesicle, u, is released into the synaptic cleft upon vesicle release.
We assumed that the area of a single spine is A, and the equations for the membrane potential at each compartment of the postsynaptic neuron are formulated as Here, V s and V pj are membrane potentials at the soma and spines, respectively.
The equations (A2)-(A3) refer to the Morris-Lecar model in which calcium and potassium currents are considered, and the postsynaptic currents are mainly involved in the dynamics of the membrane potentials at spines. In (A2), g c is the coupling conductance between the soma and the spine, and I 0 represents the integration of all possible external stimulations other than from synapses 1 to n onto the pyramidal neuron. The potassium currents are described by the ion channel opening rates n s and n pj through the equations (A4) and (A5); the opening rate of the calcium ion channel is assumed to be a constant m s,∞ . The gating variables m k,∞ , n k,∞ and τ n,k (k =s or pj (j = 1, 2, · · · , n)) are (Morris and Lecar, 1981): .
spines. The excitatory postsynaptic currents mediated by the AMPA and NMDA channels at each spine are given by whereḡ AMPA,j andḡ NMDA,j are conductances of the AMPA and NMDA channels at each spine, which are denoted as Here, g AMPA,j is the maximal conductance of the AMPA channel at each spine that is dependent on BDNF activity and given by (A15) below, and r a,j and where k bA and k dA are the basal rate of the increase and the rate of decrease of AMPA channel maximal conductance, respectively. Michaelis-Menten functions are often used in the context of enzyme kinetics (Alon, 2006).The concentration of CaMKII subunits in the postsynaptic density (PSD) is high, while the concentration of PP1 is much lower (Lisman and Zhabotinsky, 2001). Therefore, The maximal NMDA conductance g NMDA,j is dependent on the extracellular BDNF concentration ([BDNF] j ) as a result of TrkB activation following BDNF binding and was modeled according to Michaelis-Menten dynamics: [BDNF] j ). Extracellular BDNF can induce mTOR-dependent local protein synthesis in dendrites; thus, we assumed that the translation of BDNF mRNA (m BDNF for the concentration) in each dendritic spine was regulated by the level of BDNF in the synaptic cleft; the translation rate at the j'th spine is given by a function k p,j (m BDNF , [BDNF] j ) defined below. In this work, we only considered the secretion of BDNF from spine and the endocytic uptake of BDNF into the postsynaptic dendritic spines for the exchanges of BDNF proteins between cellular and synaptic cleft. We did not distinguish between post-Golgi granules and endocytosed BDNF-containing endosomes in our model, and assumed that both endogenous BDNF and endocytosed BDNF were secreted depending on the postsynaptic Ca 2+ level. These processes led to the following equations: where k d,CREB is the inactivation rate of CREB, k in and k out are the endocytosis and secretion rates of BDNF, and k dm and k dB are the degradation rates of BDNF mRNA and BDNF protein, respectively.
Michaelis-Menten kinetics is often used to non-cooperative biochemical reactions (Chen et al., 2010), and Hill type functions are often used in modeling cooperative biological processes (Gesztelyi et al., 2012;Stefan and Le Novère, 2013). In equations (A17)-(A20), the translation rate k p,j (m BDNF , [BDNF] j ) at each spine is given by a Hill type function (Alon, 2006): where k 1 is the basal translation rate independent of the cleft BDNF. The BDNF transcription rate k B ([CREB]) is formulated as a Hill type function: where k bm represents the basal transcription rate. The CREB activation rate where k b,CREB is the basal activation rate, and the dependence on [BDNF] T is provided by the Michaelis-Menten function.
The postsynaptic Ca 2+ released from the internal calcium stores include the basal release (with a rate k b,post ) and the release induced by TrkB activation (with a rate k post ([BDNF] j )) when postsynaptic TrkB receptors are bound to the cleft BDNF. In addition, the sources of postsynaptic Ca 2+ include an influx through NMDARs (with a rate k NḡNMDA,j (V Ca − V pj )). Thus, the dynamics of the postsynaptic Ca 2+ concentrations at each synapse are formulated as where τ Ca is the time constant of the Ca 2+ clearance, and the rates of BDNFinduced postsynaptic Ca 2+ release are given by a Michaelis-Menten function Default parameter values used in our study are listed in Table A1. The parameters in the equations for the membrane potential are referred to the published literature (Morris and Lecar, 1981). Other parameters are calculated to realize the three properties of LTP.