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Improve readability of Wilson-Cowan eqs in `Getting started` #8

Open gabrevaya opened 3 months ago

gabrevaya commented 3 months ago

It might look nicer (and a bit less intimidating at first glance in the getting started section) if we rearrange the equations slightly and remove the asterisk (*) for multiplication. For example, changing from the current version in the docs:


\frac{dE}{dt} = \frac{-E}{\tau_E} + \frac{1}{1 + \text{exp}(-a_E*(c_{EE}*E - c_{IE}*I - \theta_E + \eta*(\sum{jcn}))} \\[10pt] 
\frac{dI}{dt} = \frac{-I}{\tau_I} + \frac{1}{1 + exp(-a_I*(c_{EI}*E - c_{II}*I - \theta_I)}

to


\begin{align}
\nonumber
\frac{dE}{dt} &= \frac{-E}{\tau_E} + S_E(c_{EE}E - c_{IE}I + \eta\sum{jcn})\\[10pt]
\nonumber
\frac{dI}{dt} &= \frac{-I}{\tau_I} + S_I(c_{EI}E - c_{II}I)
\end{align}

with $S_E$ and $S_I$ sigmoid functions that represent the response of neuronal populations to input stimuli, defined as:

S_k(x) = \frac{1}{1 + exp(-a_kx - \theta_k)}

We could also add descriptions of what the different terms in the eqs mean. However, this would make the intro before the code a bit longer. I would include those descriptions and perhaps some more didactic intro before the equations. But maybe you would prefer to keep this as concise as possible.

hstrey commented 3 months ago

Feel free to add more explanations to the equations. It may also help to add a reference.