Synaptic Pulse extender

On Neurogrid, the synaptic conductance, $x$, is governed by
$$\tau \dot{x} = -x + g_{max}\sum_ip(t-t_i)$$
where $p(t)$ is a square pulse of length $t_{xmt}$ resulting from spikes arriving at time $t_i$, $g_{max}$ is the maximum synaptic conductance, and $\tau$ is the synaptic time constant.

Let's analyze the steady state conductance induced by a Poisson spike train where the interarrival times of spikes are distributed exponentially with pdf $f(t) = \lambda e^{-\lambda t}$.

At steady state, $\dot{x} = 0$, and $0 = -x + g_{max}p(t)$.
Therefore at steady state, $x = g_{max}p(t)$ and on average \(\langle x \rangle=g_{max} \langle p(t) \rangle\).

There is one wrinkle in our analysis: when a spike arrives within $t_{xmt}$ of the previous spike. The two resulting pulses do not add linearly.  The second pulse merely extends the previous pulse by the time between the spikes.

$$\langle p(t)\rangle=\langle \mathrm{spike\ rate}\rangle \langle \mathrm{average\ pulse\ value}\rangle$$

For a Poisson process, the rate is simply $\lambda$.

For a full pulse, the area is simply $t_{xmt}$.  For the pulse resulting from a collision, the area is $t=\Delta t$. So

$$\langle \mathrm{average\ pulse\ value}\rangle = \int_0^{t_{xmt}} t \lambda e^{-\lambda t} dt + \int_{t_{xmt}}^\infty t_{xmt}\lambda e^{-\lambda t} dt$$

$$= \left. t e^{-\lambda t}\right|_{t_{xmt}}^0 + \int_0^{t_{xmt}} e^{-\lambda t} dt + \left. t_{xmt}\lambda e^{-\lambda t} \right|_\infty^{t_{xmt}}$$
$$= -t_{xmt} e^{-\lambda t_{xmt}} + \left. \frac{1}{\lambda} e^{-\lambda t} \right|_{t_{xmt}}^0 + t_{xmt}\lambda e^{-\lambda t_{xmt}}$$

$$= \frac{1}{\lambda} (1-e^{-\lambda t_{xmt}})$$

$$\langle p(t)\rangle = \lambda \frac{1}{\lambda} (1-e^{-\lambda t_{xmt}})$$
$$\langle p(t)\rangle = (1-e^{-\lambda t_{xmt}})$$

The average synaptic conductance is then
$$\langle x \rangle = g_{max}(1-e^{-\lambda t_{xmt}})$$