Computational Examination of Synaptic Plasticity and Metaplasticity in Hippocampal Dentate Granule Neurons

Abstract

Long-term potentiation (LTP) and long-term depression (LTD) are two forms of long-lasting synaptic plasticity. To protect synaptic weights from extreme increase or decrease, neurons need to regulate their activities; this phenomenon is called homeostatic plasticity. The induction of homosynaptic plasticity by high-frequency stimulation (HFS) increases the strength of synaptic weights dramatically which makes a neuron loses balance. However, heterosynaptic plasticity keeps the synaptic weights away from the extreme increase and brings them into a stable range. Therefore, neurons need both homosynaptic and heterosynaptic plasticity to regulate their synaptic weights. In most previous studies of spike-timing-dependent plasticity (STDP) models, postsynaptic spikes are treated as all-or-none events; however, in this study, we calculate the voltage of the postsynaptic spikes instead of counting the number of spikes. Further, we incorporate a modified model of metaplasticity based on the voltage of the spike rather than the spike count. To model synaptic plasticity of dentate granule cells, we used computational simulations and employed STDP rules accompanied with metaplasticity model and noisy spontaneous activity to address these questions; firstly, could our plasticity and metaplasticity models produce homosynaptic LTP in one pathway and heterosynaptic LTD in the neighbouring pathway? Secondly, does the magnitude of spontaneous activity after stimulation determine the level of heterosynaptic LTD? Thirdly, when two stimulations with the same frequency are applied to the same synapse at different time interval, will both stimulations produce the same level of synaptic plasticity? Our result shows that employing STDP and metaplasticity rules based on the voltage of the spikes accompanied with noisy spontaneous activity could replicate homosynaptic LTP in the stimulated pathway and heterosynaptic LTD in the non-stimulated neighbouring pathway of the dentate granule cell, as shown experimentally (Abraham WC, Mason-Parker SE, Bear MF, Webb S, Tate WP, Proc Natl Acad Sci 98(19):10924–10929, 2001; Abraham WC, Logan B, Wolff A, Benuskova L, J Neurophysiol 98(2):1048–1051, 2007).

Appendix

Integration Method of Crank-Nicolson

The Crank-Nicolson method or the central difference is a finite difference method with the accuracy of the second order in time, which was developed by Crank and Nicolson. The error oscillation of this method decays with time; therefore the solution is stable and safe for most solutions. The Crank-Nicolson method is a combination of the backward and forward Euler methods. It is equivalent to advancing by one-half step using backward Euler and then advancing by one-half step using forward Euler. The global error for this method is proportional to the square of the step size (Hines and Carnevale 1997).

By considering the forward Euler method as:

$$ \frac{y_{n+1}-{y}_n}{\Delta t}=f\left({t}_n,{y}_n\right) $$

(A.1)

and the backward Euler method as:

$$ \frac{y_{n+1}-{y}_n}{\Delta t}=f\left({t}_{n+1},{y}_{n+1}\right) $$

(A.2)

Therefor the Crank-Nicolson method can be calculated as:

$$ \frac{y_{n+1}-{y}_n}{\Delta t}=\frac{f\left({t}_n,{y}_n\right)+f\left({t}_{n+1},{y}_{n+1}\right)}{2} $$

(A.3)

The RC Circuit

With injecting the current into the circuit, the membrane potential changes. Therefore, Kirchhoff’s current law indicates that the total current entering into the junction is equal to the total current leaving the junction. The summation of the membrane current Ia and injected current I _e is equal to the summation of the capacitance current I _c a and ionic current I _i a. a is the curved surface area of the cylinder (Sterratt et al. 2011).

$$ Ia+{I}_e={I}_ca+{I}_ia $$

(A.4)

The following equation shows the ionic current flows through the resistor:

$$ {I}_ia=\frac{V-{E}_m}{\frac{R_m}{a}} $$

(A.5)

where $\frac{R_m}{a}$ is the membrane resistance and E _m is the equilibrium potential of the membrane. According to the following equation, the capacitive current is proportional to the rate of change of the voltage:

$$ {I}_ca={C}_ma\frac{\mathrm{d}V}{\mathrm{d}t} $$

(A.6)

The membrane capacitance is C _m a. If we suppose that the circuit is isolated, then Ia is zero. With substituting the I _i and I _c in Eq. 4, we have:

$$ {C}_m\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{E_m+V}{R_m}+\frac{I_e}{a} $$

(A.7)

This is the first-order ordinary differential equation (ODE) for the membrane potential V with units from Table 5 (Sterratt et al. 2011).

Table 5 Passive quantities

Multi-compartmental Models

Multi-compartmental models are useful techniques to model dendritic trees of the neuron. In this model, dendritic tree breaks up into the small compartment. With considering each compartment as a cylinder with a length l and a diameter d, the surface area will be equal to a = πdl (Ermentrout and Terman 2010). Current flows through each compartment into the membrane capacitance and the membrane resistance. It also flows through intracellular and extracellular of the membrane and can be modelled by axial resistances. The extracellular resistance can be considered zero. R _a is the specific axial resistance with units Ω cm, and the axial resistance of the cylindrical compartment is 4R _a l/πd ² in which πd ²/4 is a cross-sectional area. j is the number of compartment, and V _j is the membrane potential in the j the compartment, and I _e,j is the injected current into the compartment j. The membrane current I _j is equal to the sum of the leftwards and rightwards axial currents. Therefore, according to Ohm’s law:

$$ {I}_ja=\frac{V_{j+1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.}+\frac{V_{j-1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.} $$

(A.8)

where I _j a is a current, and according Kirchhoff’s current law:

$$ {I}_{c,j}a+{I}_{i,j}a={I}_ja+{I}_{e,j}\vspace*{-3pt} $$

(A.9)

$$ {I}_{c,j}a+{I}_{i,j}a=\frac{V_{j+1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.}+\frac{V_{j-1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.}+{I}_{e,j} $$

(A.10)

The following equations are similar to Eq. 10 for a patch of membrane with two extra equations which describe the flowing current through two compartments j − 1 and j + 1:

$$ \pi dl{C}_m\frac{\mathrm{d}{V}_j}{\mathrm{d}t}=\frac{E_m-{V}_j}{\raisebox{1ex}{${R}_m$}\!\left/ \!\raisebox{-1ex}{$\pi dl$}\right.}+\frac{V_{j+1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.}+\frac{V_{j-1}-{V}_j}{\raisebox{1ex}{$4{R}_al$}\!\left/ \!\raisebox{-1ex}{$\pi {d}^2$}\right.}+{I}_{e,j} $$

(A.11)

Where a is the surface area of the cylinder.

$$ {C}_m\frac{\mathrm{d}{V}_j}{\mathrm{d}t}=\frac{E_m-{V}_j}{R_m}+\frac{d}{4{R}_a}\left(\frac{V_{j+1}-{V}_j}{l^2}+\frac{V_{j-1}-{V}_j}{l^2}\right)+\frac{I_{e,j}}{\pi dl} $$

(A.12)

This equation is the fundamental equation for the compartmental model (Sterratt etal. 2011).

Rate Functions for Nine-Compartmental Models of GC

Activation and inactivation gates at compartment i are formulated as:

$$ \frac{\mathrm{d}{z}_i}{\mathrm{d}t}={\alpha}_{{z_i}_i}-\left({\alpha}_{z_i}+{\beta}_{z_i}\right){z}_i\vspace*{-4pt} $$

(A.13)

$$ \left({z}_i:{m}_i,{h}_i,{n}_{f,i},{n}_{s,i},{k}_i,{l}_i,{a}_i,{b}_i,{c}_i,{d}_i,{e}_i,{r}_i,{q}_i\ \right) $$

variable z _i represents m _i, h _i, n _{f, i}, n _{s, i}, k _i, l _i, a _i, b _i, c _i, d _i, e _i, r _i and q _i ion-gating variables. Rate functions at compartment i determine the transition between open and closed states of the ion channels. The following equations show the rate functions at compartment i (Aradi and Holmes 1999):

$$ {\alpha}_{m,i}(V)=\frac{-0.3\left({V}_i-25\right)}{\left[\exp \left(\frac{V_i-25}{-5}\right)-1\right]} $$

(A.14)

$$ {\beta}_{m,i}(V)=\frac{0.3\left({V}_i-53\right)}{\left[\exp \left(\frac{V_i-53}{5}\right)-1\right]} $$

(A.15)

$$ {\alpha}_{h,i}(V)=\frac{0.23}{\exp \left(\frac{V_i-3}{20}\right)} $$

(A.16)

$$ {\beta}_{h,i}(V)=\frac{3.33}{\left[\exp \left(\frac{V_i-55.5}{-10}\right)+1\right]} $$

(A.17)

$$ {\alpha}_{n_f,i}(V)=\frac{-0.07\left({V}_i-47\right)}{\left[\exp \left(\frac{V_i-47}{-6}\right)-1\right]} $$

(A.18)

$$ {\beta}_{n_f,i}(V)=\frac{0.264}{\exp \left(\frac{V_i-22}{40}\right)} $$

(A.19)

$$ {\alpha}_{n_s,i}(V)=\frac{-0.028\left({V}_i-35\right)}{\left[\exp \left(\frac{V_i-35}{-6}\right)-1\right]} $$

(A.20)

$$ {\beta}_{n_s,i}(V)=\frac{0.1056}{\exp \left(\frac{V_i-10}{40}\right)} $$

(A.21)

$$ {\alpha}_{k,i}(V)=\frac{-0.05\left({V}_i+25\right)}{\left[\exp \left(\frac{V_i+25}{-5}\right)-1\right]} $$

(A.22)

$$ {\beta}_{k,i}(V)=\frac{0.1\left({V}_i+15\right)}{\left[\exp \left(\frac{V_i+15}{8}\right)-1\right]} $$

(A.23)

$$ {\alpha}_{l,i}(V)=\frac{0.00015}{\exp \left(\frac{V_i+13}{15}\right)} $$

(A.24)

$$ {\beta}_{l,i}(V)=\frac{0.06}{\left[\exp \left(\frac{V_i+68}{-12}\right)+1\right]} $$

(A.25)

$$ {\alpha}_{a,i}(V)=\frac{0.2\left(19.26-{V}_i\right)}{\left[\exp \left(\frac{19.26-{V}_i}{10}\right)-1\right]} $$

(A.26)

$$ {\beta}_{a,i}(V)=0.009\ \exp \left(\frac{-{V}_i}{22.03}\right) $$

(A.27)

$$ {\alpha}_{b,i}(V)={10}^{-6}\exp \left(\frac{-{V}_i}{16.26}\right) $$

(A.28)

$$ {\beta}_{b,i}(V)=\frac{1}{\left[\exp \left(\frac{29.76-{V}_i}{10}\right)+1\right]} $$

(A.29)

$$ {\alpha}_{c,i}(V)=\frac{0.19\left(19.88-{V}_i\right)}{\left[\exp \left(\frac{19.88-V}{10}\right)-1\right]} $$

(A.30)

$$ {\beta}_{c,i}(V)=0.046\ \exp \left(\frac{-{V}_i}{20.76}\right) $$

(A.31)

$$ {\alpha}_{d,i}(V)=1.6\times {10}^{-4}\exp \left(\frac{-{V}_i}{148.4}\right) $$

(A.32)

$$ {\beta}_{d,i}(V)=\frac{1}{\left[\exp \left(\frac{39-{V}_i}{10}\right)+1\right]} $$

(A.33)

$$ {\alpha}_{e,i}(V)=\frac{15.69\ \left(81.5-{V}_i\right)}{\left[\exp \left(\frac{81.5-{V}_i}{10}\right)-1\right]} $$

(A.34)

$$ {\beta}_{e,i}(V)=0.29\ \exp \left(\frac{-{V}_i}{10.86}\right) $$

(A.35)