Plausible Neural Networks?

Mastebroek and Vos (eds) "Plausible Neural Networks for Biological Modelling", 2001.
The first part of the book is concerned more with the empirical and theoretical analysis of neurons that can be used for the construction of the actual neural networks that are presented in the second part.

"Integrate and Fire Model"
in ch,2 Gerstner seems to have anticipated my challenge to show how neurons can perform mathematical operations. He even has a very nice diagram in two parts (fig.2,4, p.29). One part showing two neurons synapsing with each other. The other part is of course the most important. It translates neuronal functions in formal devices that compute specific parts of a function.
The mathematical functions, (formulas 2.6 to 2.8) are not really important as such. What matters is the fact they can be put into a clear diagram containing a resistance element R, a capacitance C and the necessary voltage, input and output symbols. Because there is a clear link with certain biological properties of neurons (membrane resistance, action potential, etc), the diagram seems to possess a high degree of biological plausibility.
Let us look at it more closely.
We will probably notice right away that all the processes described are of an electrical nature. Which should not be a problem. After all, when describing the electrical properties of a circuit, we do not need to specify the kind of molecules a battery contains. Its voltage and resistance, and other quantitative figures, tell us all we need to know.
But such a diagram has bigger pretensions. It aims not only at explaining electrical phenomena within and between neurons, but also at providing a clear image of the behavior of a neuron when it it gets different inputs from multiple neurons: the integrating part, that again, deserves it own formulas or equations (2.9 to 2.11).
What could that possibly mean?

One neuron getting multiple input and reacting in a certain way. 
Let us not forget that a neuronal input produces, directly or indirectly, either a sensory sensation, a memory, thought or emotion, or a bodily reaction (visceral or muscular). The same holds for any neuronal output.
The equations that our author overwhelms us with treat all neurons indifferently, as random elements whose value is determined by the equations themselves.
Even if the electrical analysis is right, it still does not tell us anything worth knowing concerning the functions specific neurons are fulfilling when reacting exactly as predicted by the formulas. Which means that we are in fact facing completely useless computations that do not help us in any way at better understanding what is happening in the brain.
Furthermore, nor the diagram nor the formulas relate really to biological neurons.

The diagram is that of an electrical circuit that can produce certain electrical results. What would be more interesting is showing how the biological processes in a neuron fulfill all the functions described in the diagram. But all the author has done is translate his verbal description into a more scientific sounding diagram and formulas. Here is his initial description:
"A neuron is surrounded by its cell membrane. Ions may pass through the membrane at pores or specific channels which may be open or closed. A rather simple picture of the electrical properties of a cell is the following.
Close to the inactive rest state the neuron is characterized by some resistance R in parallel with some capacitance C. The factor RC= [some squiggles] defines the membrane time constant of the neuron. the voltage u will be measured with respect to the neuronal resting potential. If the neuron is stimulated by some current I, the voltage u rises according to [formula 2.6]."
As you can see, the behavior and properties of neurons have been brought down to those of elements in an electrical circuit. And that is all the diagram does and can show.
Getting to the integrating ability of neurons, the description changes accordingly:
"In a real cortical network the driving current is the synaptic input which arises as a result of the arrival of spikes from other neurons. Let us suppose that a spike of a presynaptic neuron j which was fired at time [squiggles] evokes some current [squiggles] at the synapse connecting neuron j to neuron i. The factor [squiggles] determines the amplitude of the current pulse and will be called the synaptic efficacy [ah! That one!]. The function [squiggles] describes the time course of the synaptic current. If neuron i receives input from several presynaptic neurons j, the total input current to neuron i is [formula 2.9]."
Again, nothing in this description goes beyond that of an electrical circuit, and the mention of "a real cortical network" is in no way justified.

As is often the case, a neural process is interpreted in such a way as to make the use of mathematic formulas possible, and then those same formulas are used as the indisputable proofs of the plausibility of the explanation.
Whatever Gerstner did with his voodoo incantations (all the beautiful equations), he certainly did not explain how neurons could perform the mathematical integration of multiple inputs, nor did he give us a clue as to what these electrical currents could possibly mean to a functioning brain.