$\tau \dot{V}=-V(t)+I(t)$ ...Eqn(1)
[Voltage and current have same units.]
Synaptic efficacies (voltage units) are assumed to be distributed as $P(J)$ with mean $J$ and standard deviation $\Delta J$. Synaptic current follows the dynamics (Each synapse is a single exponential decaying synapse):
$\tau' \dot{I}=-I(t) + \sum_i J_i \tau \sum_k \delta(t-t_i^k)$ ...Eqn(2)
[$J_i$ has voltage units, $\delta$ function has 1/time units.]
[Why should membrane time constant appear here? Ideally it is $V-V_{reversal}$ that appears. $V$ has dynamics at time scales of $\tau$, so there is some justification...]
If $\tau'<<\tau$, then
$\tau\dot{V}=-V(t)+\sum_i J_i\tau\sum_k\delta(t-t_i^k)$ ...Eqn(3)
Thus, integrating a single synaptic event at $t=0$, we get an EPSP:
$V(t)=V(0)\exp(-t/\tau)+J_i\exp(-t/\tau)$ for $t>0$.
Assume low input rate $\nu$, but many synaptic inputs $C$ and long enough interval $\tau$, such that number of incoming spikes are high in interval $\tau$. Then the summed synaptic current $I(t)$ over interval $\tau$ has a Gaussian distribution with mean $\mu(\nu)$ and SD $\sigma(\nu)$.
Assume, no decay of membrane potential in interval $\tau$, then with threshold $\theta$, the probability of a spike occurring in interval $\tau$ is:
$Pr(\nu) = \int_\theta^\infty \frac{dI}{\sqrt{2\pi\sigma^2(\nu)}} \exp\left(-\frac{\left(I-\mu(\nu)\right)^2}{2\sigma^2(\nu)}\right)$ ...Eqn(4)
[I think this assumes Gaussian white noise, i.e. all possible values are attained in interval $\tau$. What about the Gaussian noise
generated algorithmically sampled at $\Delta t$.]
The resulting output spike rate is $\nu_{out}=Pr(\nu)/\tau$ ...Eqn(6).
look up mean first passage time derivations...
For the simplified IF neuron assuming no decay in interval $\tau$, we have the mean and SD of the summed current over interval $\tau$ as:
$\mu(\nu) = JC\nu\tau$, where $C$ are the number of synapses; $\sigma(\nu) = J\sqrt{C\nu\tau(1+\Delta^2)}$ ...Eqn(8).
Note that $\Delta J$ is the variance in the synaptic efficacies, and there is a variance due to Poisson spikes.
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