# What's a reasonable time constant for a mathematical model of a neuron?

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?

• How these parameters relate to your firing rates? Neurons have a maximum firing rate of 500-1000 Hz. – BPinto Mar 22 at 3:00