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
\[\langle x \rangle = g_{max}(1-e^{-\lambda t_{xmt}})\]