I am trying to implement a Biexponential shape conductance base synapse:

I_{syn} = g_{syn} (V-V_{ref})\sum_j s_j \\
\dot{s_j} = \alpha H_{\infty}(v_j-\theta) (1-s) - \beta s\\
H_{\infty} = \frac{1}{1+\exp(-(v-\theta^H)/\sigma^H)}

\theta, \theta_H and \sigma are constants. Adding H_{\infty} is the question.

Following the documentation I came up to this :

I skipped the H_{\infty} for now:

I think this should be :

g_{syn}(t) = g_{peak} \frac{e^{-t/\tau_1}-e^{-t/\tau_2}}{e^{-t_{peak}/\tau_1}-e^{-t_{peak}/\tau_2}} \Theta(t)\\
t_{peak} = \frac{\tau_1 \tau_2}{\tau_2 - \tau_1} \ln{\frac{\tau_2}{\tau_1}}

\Theta is Heaviside step function. But I used the documentation method for the following.

```
tau_1 = 1 / alpha
tau_2 = 1 / beta
scale_f = (tau_2 / tau_1) ** (tau_1 / (tau_2 - tau_1))
eqs_e = """
VT : volt
IX : amp
Im = IX +
gL * (EL - vm) +
gL * DeltaT * exp((vm - VT) / DeltaT) : amp
ds/dt = -s / Tau_d : siemens
dg_syn/dt = scale_f * (s - g_syn) / Tau_r : siemens
I_syn = g_syn * (Erev - vm): amp
dvm/dt = (Im + I_syn) / C : volt
"""
```

for two neuron and a connection from 1 to two, the neuron 1 get and input current and I record `g_syn`

and `I_syn`

from the second neuron. `tau_1 = 2ms, tau_2=10ms`

Am I right up to now?

The question is how to include

H_{\infty}?

or adding H is just complicating and we can safely approximate it.

Hi. Your equations look correct to me, except maybe for the normalization (`Tau_r`

needs to be `tau_1`

and `Tau_d`

needs to be `tau_2`

I think, but maybe that’s already the case).

Regarding the H_\infty function: there are two different ways to describe synapses. One approach is to describe them as a PSP (or PSC) triggered by an instantaneous event, the pre-synaptic spike (in equations, this is often written as an expression involving the Dirac delta \delta(t - t_\mathrm{spike})). This is the most common way to describe spikes in Brian and this is what you used here by having something like `on_pre='s += w'`

in your `Synapses`

description. The other approach is to describe everything as a continous function of the pre-synaptic membrane potential, i.e. without any reference to an instantanous spike that triggers the synaptic event. This takes a form like the equations you cite in the beginning, where H_\infty refers to the *pre-synaptic* membrane potential. Typically, the parameters (in your equations \theta^H and \sigma^H) are chosen in a way that if you plot H_\infty for pre-synaptic action potential waveform, you will find that it more or less looks like an instantaneous pulse, i.e. close to a Dirac delta. I would say that you need this more complex formulation only under special circumstances, e.g. when modeling graded synaptic transmission where the effect of a spike depends on the pre-synaptic waveform. There are two main disadvantages to this formulation: 1) it is computationally much more demanding, since you have to update the synaptic conductances at every time step for each synapse, even in the time steps where the pre-synaptic cell did not spike (i.e. where H_\infty is basically 0) and 2) it requires your pre-synaptic model to have a somewhat realistic representation of an action potential. This excludes most integrate-and-fire model. You might get away with using an EIF as in your equations, but I would not recommend it as the exact value of H_\infty for a spike will depend on timing details.

In general, an approach for the continuous coupling approach would make used of summed variables to update the PSP at every time step.

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Very informative.

You are right, The synapse equation come from Terman 2002 for STN-GPe circut and neuron model is HH type and I use EIF model for simplicity.

Just to share something if some one has any idea:

Blow figure are the time course for simple form and including H_{\infty} in HH type neurons. One thing that is noticeable in `I_syn`

is that we have negative value which I think is coming from `v-v_ref`

in synapse term:

I_{syn} = g_{syn} (V-V_{ref})\sum_j s_j

it is sometimes negative. I ask from some one and he believed this a common mistake among people and we need to consider a constant value (constant negative or positive) instead of this term.

Please let me know if I am wrong or there is better idea about it.

Not sure whether this is still relevant, but I do not think this is a mistake nor that V-V_{ref} should be replaced by a constant term. If you replaced it by a constant term, you’d change a conductance-based synapse model to a current-based model, i.e. use a simpler approximation to the biophyical system. Of course, you could do that for performance reasons in a big network or just generally for simplicity, but it would be a less precise model. The actual current that is flowing in real neurons *is* dependent on the voltage potential. For excitatory synapses this is not that relevant most of the time, since the membrane potential is usually very far from the V_{ref}. For inhibitory synapses, this can make quite a bit of a difference though (cf. shunting).

I see that this looks odd during an action potential, but this is a bit of an extreme situation. I think any synaptic currents should be small compared to the sodium and potassium currents during a spike, so that an excitatory synapse has an inhibitory effect should not matter much.

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