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I am trying to understand an equation from the textbook "Theory of neural information processing systems" by Coolen, Sullich and Kuhn.

The book states that "the evolution in time of the postsynpatic potential V(t) can be written as a linear differential equation of the form:

[tex]\frac {d}{dt}V(t) = \frac {d}{dt}V(t) |_{passive} + \frac {d}{dt}V(t) |_{reset}[/tex]"

The book then addreses the passive component:

"The first term represents the passive cable-like electric behaviour of the dendrite:

[tex]\tau \frac{d}{dt}V(t) |_{passive} = \frac{1}{\rho} \left[ \tilde{I} + \sum_{k = 1}^{N}I_{k}(t) - \rho V(t) \right][/tex]

In which the two parameters [itex]\tau[/itex] and [itex]\rho[/itex] reflect the electrical properties of the dendrite ([itex]\tau[/itex] being the characteristic time for current changes to affect the voltage, [itex]\rho[/itex] controlling the stationary ratio between voltage and current), and [itex]\tilde{I}[/itex] represents the stationary currents due to the ion pumps."

The summation of I

_{k}(t) is the contribution of currents at the synapses with sending neurons.

I'm already confused. I am not sure what it is actually claiming these parameters and so forth are; if [itex]\tilde{I}[/itex] "represents the stationary current due to the ion pumps", then I take this to mean that these ion pumps work continuously at a set pace, never changing the rate at which they move ions across the membrane, regardless of what else is happening; a stationary current is time-independent. In this case, [itex]\tilde{I}[/itex] is a constant, and it has units of current. [itex]\tau[/itex] and [itex]\rho[/itex] are equally confusing; I think [itex]\tau[/itex] has units of time, and [itex]\rho[/itex] has units of current over voltage. Take the case, given in the textbook, where there are no contributions from other neurons; in this case, I

_{k}(t) = 0, and :

[tex]\tau \frac{d}{dt}V(t) |_{passive} = \frac{\tilde{I}}{\rho} - V(t)[/tex]

The textbook claims that [itex]\frac{\tilde{I}}{\rho} = V_{rest}[/itex]. The resting membrane potential is, essentially, a constant. If both [itex]\tilde{I}[/itex] and V

_{rest}are constants, then [itex]\rho[/itex] is also a constant. This is fine, since [itex]\rho[/itex] is controlling "the stationary ratio between voltage and current". What we then have is:

[tex] \frac{d}{dt}V(t) = \frac{V_{rest} - V(t)}{\tau}[/tex]

This equation makes no sense to me; I don't see what it is trying to show. The units work out fine at all steps, but I am convinced that what I think the parameters are doing is completely wrong. [itex]\tilde{I}[/itex] for example, if it represents a time-independent current, and it doesn't appear to be a function of something else, it is then a constant, but for a membrane at rest, the net current has to be zero; V

_{rest}is generally not zero, and so [itex]\tilde{I}[/itex] cannot be zero, it is not then the summation of all the ions pumps 'doing their thing' when the membrane is at rest. The same for the other parameters, I do not see where they come from, or what they actually represent; I think [itex]\tau[/itex] is connected to capacitance, but I am grasping at straws, I think.

Sorry that this is a little unstructured. I wanted to include my "attempt" to piece together what is happening. Basically, if anyone can give any guidance on the reasoning behind this equation, that would be much appreciated.