Doc fix and standardize Dual Exponential model again (#591)

* [doc] update documentation

* update

README.md

@@ -104,4 +104,6 @@ We also welcome your contributions
 - [ ] pipeline parallelization on multiple devices for sparse spiking network models
 - [ ] multi-compartment modeling
 - [ ] measurements, analysis, and visualization methods for large-scale spiking data
+- [ ] Online learning methods for large-scale spiking network models
+- [ ] Classical plasticity rules for large-scale spiking network models
 

brainpy/_src/dyn/_docs.py

@@ -40,3 +40,166 @@
 
 ltc_doc = 'with liquid time-constant'
 
+
+dual_exp_syn_doc = r'''
+
+  **Model Descriptions**
+
+  The dual exponential synapse model [1]_, also named as *difference of two exponentials* model,
+  is given by:
+
+  .. math::
+
+    g_{\mathrm{syn}}(t)=g_{\mathrm{max}} A \left(\exp \left(-\frac{t-t_{0}}{\tau_{1}}\right)
+        -\exp \left(-\frac{t-t_{0}}{\tau_{2}}\right)\right)
+
+  where :math:`\tau_1` is the time constant of the decay phase, :math:`\tau_2`
+  is the time constant of the rise phase, :math:`t_0` is the time of the pre-synaptic
+  spike, :math:`g_{\mathrm{max}}` is the maximal conductance.
+
+  However, in practice, this formula is hard to implement. The equivalent solution is
+  two coupled linear differential equations [2]_:
+
+  .. math::
+
+      \begin{aligned}
+      &\frac{d g}{d t}=-\frac{g}{\tau_{\mathrm{decay}}}+h \\
+      &\frac{d h}{d t}=-\frac{h}{\tau_{\text {rise }}}+ (\frac{1}{\tau_{\text{rise}}} - \frac{1}{\tau_{\text{decay}}}) A \delta\left(t_{0}-t\right),
+      \end{aligned}
+
+  By default, :math:`A` has the following value:
+
+  .. math::
+
+     A = \frac{{\tau }_{decay}}{{\tau }_{decay}-{\tau }_{rise}}{\left(\frac{{\tau }_{rise}}{{\tau }_{decay}}\right)}^{\frac{{\tau }_{rise}}{{\tau }_{rise}-{\tau }_{decay}}}
+
+  .. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
+         "The Synapse." Principles of Computational Modelling in Neuroscience.
+         Cambridge: Cambridge UP, 2011. 172-95. Print.
+  .. [2] Roth, A., & Van Rossum, M. C. W. (2009). Modeling Synapses. Computational
+         Modeling Methods for Neuroscientists.
+  
+'''
+
+dual_exp_args = '''
+    tau_decay: float, ArrayArray, Callable. The time constant of the synaptic decay phase. [ms]
+    tau_rise: float, ArrayArray, Callable. The time constant of the synaptic rise phase. [ms]
+    A: float. The normalization factor. Default None.
+
+'''
+
+
+alpha_syn_doc = r'''
+
+  **Model Descriptions**
+
+  The analytical expression of alpha synapse is given by:
+
+  .. math::
+
+      g_{syn}(t)= g_{max} \frac{t-t_{s}}{\tau} \exp \left(-\frac{t-t_{s}}{\tau}\right).
+
+  While, this equation is hard to implement. So, let's try to convert it into the
+  differential forms:
+
+  .. math::
+
+      \begin{aligned}
+      &\frac{d g}{d t}=-\frac{g}{\tau}+\frac{h}{\tau} \\
+      &\frac{d h}{d t}=-\frac{h}{\tau}+\delta\left(t_{0}-t\right)
+      \end{aligned}
+  
+  .. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
+        "The Synapse." Principles of Computational Modelling in Neuroscience.
+        Cambridge: Cambridge UP, 2011. 172-95. Print.
+
+
+'''
+
+
+exp_syn_doc = r'''
+
+  **Model Descriptions**
+
+  The single exponential decay synapse model assumes the release of neurotransmitter,
+  its diffusion across the cleft, the receptor binding, and channel opening all happen
+  very quickly, so that the channels instantaneously jump from the closed to the open state.
+  Therefore, its expression is given by
+
+  .. math::
+
+      g_{\mathrm{syn}}(t)=g_{\mathrm{max}} e^{-\left(t-t_{0}\right) / \tau}
+
+  where :math:`\tau_{delay}` is the time constant of the synaptic state decay,
+  :math:`t_0` is the time of the pre-synaptic spike,
+  :math:`g_{\mathrm{max}}` is the maximal conductance.
+
+  Accordingly, the differential form of the exponential synapse is given by
+
+  .. math::
+
+      \begin{aligned}
+       & \frac{d g}{d t} = -\frac{g}{\tau_{decay}}+\sum_{k} \delta(t-t_{j}^{k}).
+       \end{aligned}
+
+  .. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
+          "The Synapse." Principles of Computational Modelling in Neuroscience.
+          Cambridge: Cambridge UP, 2011. 172-95. Print.
+
+'''
+
+
+std_doc = r'''
+
+  This model filters the synaptic current by the following equation:
+
+  .. math::
+
+     I_{syn}^+(t) = I_{syn}^-(t) * x
+
+  where :math:`x` is the normalized variable between 0 and 1, and
+  :math:`I_{syn}^-(t)` and :math:`I_{syn}^+(t)` are the synaptic currents before
+  and after STD filtering.
+
+  Moreover, :math:`x` is updated according to the dynamics of:
+
+  .. math::
+
+     \frac{dx}{dt} = \frac{1-x}{\tau} - U * x * \delta(t-t_{spike})
+
+  where :math:`U` is the fraction of resources used per action potential,
+  :math:`\tau` is the time constant of recovery of the synaptic vesicles.
+
+'''
+
+
+stp_doc = r'''
+
+  This model filters the synaptic currents according to two variables: :math:`u` and :math:`x`.
+
+  .. math::
+
+     I_{syn}^+(t) = I_{syn}^-(t) * x * u
+
+  where :math:`I_{syn}^-(t)` and :math:`I_{syn}^+(t)` are the synaptic currents before
+  and after STP filtering, :math:`x` denotes the fraction of resources that remain available
+  after neurotransmitter depletion, and :math:`u` represents the fraction of available
+  resources ready for use (release probability).
+
+  The dynamics of :math:`u` and :math:`x` are governed by
+
+  .. math::
+
+     \begin{aligned}
+     \frac{du}{dt} & = & -\frac{u}{\tau_f}+U(1-u^-)\delta(t-t_{sp}), \\
+     \frac{dx}{dt} & = & \frac{1-x}{\tau_d}-u^+x^-\delta(t-t_{sp}), \\
+     \end{aligned}
+
+  where :math:`t_{sp}` denotes the spike time and :math:`U` is the increment
+  of :math:`u` produced by a spike. :math:`u^-, x^-` are the corresponding
+  variables just before the arrival of the spike, and :math:`u^+`
+  refers to the moment just after the spike.
+
+
+'''
+