Background I've been studying various mathematical models of neurons. So far I've covered the classic Hodgkin-Huxley model (to describe the potential difference of a single neuron) and the integrate and fire model for two neurons (to describe the voltages of interacting neurons). I am now studying a firing-rate model which describes the firing rate of neurons, not their voltage.
All of these models include a time constant $\tau$. For the Hodgkin-Huxley and integrate-and-fire models $\tau$ tells us how fast the channels and pumps open/close in response to an action potential.
I am currently working with a two-neuron system of one excitatory neuron and one inhibitory neuron. It is described by the equations
$\tau_E \frac{dV}{dt} = f(v_E, v_I, \theta)$
$\tau_I \frac{dV}{dt} = g(v_E, v_I, \theta)$
where
- $v_E$ is the firing rate of the excitatory neuron
- $v_I$ is the firing rate of the inhibitory neuron
- $\theta$ is a set of constant parameters.
- $f$ and $g$ are linear WRT $v_E$ and $v_I$
I have fixed all parameters $\tau_E$ and $\theta$ leaving only $\tau_I$ to be varied. Certain values of $\tau_I$ cause stability while others cause instability.
For the parameters $\theta$ that I'm using I have found that disregarding the possibilities of $\tau_I<8$ and $\tau_I>791$ greatly simplifies the analysis (because of the way the eigenvalues of the Jacobian matrix work out). I also have reason to believe that it might be alright to disregard these values.
All of the $\tau$ time constants in all the examples I've seen so far have been around $10ms$.
Question $791$ seems very high, but is it so biologically unreasonable that it can be completely ignored? What's to stop one neuron being $1000$ times more responsive than another one given there's such diversity in neuronal structure and function over all types of neurons? Are there examples of this?
$8$ seems completely reasonable for a time constant. Surely values under $8$ can't be disregarded completely?
